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Riemann hypothesis
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Riemann hypothesis , in number theory , hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function , which is connected to the prime number theorem and has important implications for the distribution of prime numbers . Riemann included the hypothesis in a paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (“On the Number of Prime Numbers Less Than a Given Quantity”), published in the November 1859 edition of Monatsberichte der Berliner Akademie (“Monthly Review of the Berlin Academy”).
Riemann extended the study of the zeta function to include the complex numbers x + i y , where i = Square root of √ −1 , except for the line x = 1 in the complex plane. Riemann knew that the zeta function equals zero for all negative even integers −2, −4, −6,… (so-called trivial zeros) and that it has an infinite number of zeros in the critical strip of complex numbers that fall strictly between the lines x = 0 and x = 1. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1 / 2 . Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.
In 1914 English mathematician Godfrey Harold Hardy proved that an infinite number of solutions of ζ( s ) = 0 exist on the critical line x = 1 / 2 . Subsequently it was shown by various mathematicians that a large proportion of the solutions must lie on the critical line, though the frequent “proofs” that all the nontrivial solutions are on it have been flawed. Computers have also been used to test solutions, with the first 10 trillion nontrivial solutions shown to lie on the critical line.
A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography .
The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics . It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated Hilbert’s idea with a list of important problems for the 21st century; the Riemann hypothesis was number one. In 2000 it was designated a Millennium Problem , one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., U.S., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency ( DARPA ) listed it as one of the DARPA Mathematical Challenges, 23 mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Nineteen: Settle the Riemann Hypothesis. The Holy Grail of number theory.”
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Science News
Here’s why we care about attempts to prove the riemann hypothesis.
The latest effort shines a spotlight on an enduring prime numbers mystery
LINED UP The Riemann zeta function has an infinite number of points where the function’s value is zero, located at the whirls of color in this plot. The Riemann hypothesis predicts that certain zeros lie along a single line, which is horizontal in this image, where the colorful bands meet the red.
Empetrisor/Wikimedia Commons ( CC BY-SA 4.0 )
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By Emily Conover
September 25, 2018 at 11:46 am
A famed mathematical enigma is once again in the spotlight.
The Riemann hypothesis, posited in 1859 by German mathematician Bernhard Riemann, is one of the biggest unsolved puzzles in mathematics. The hypothesis, which could unlock the mysteries of prime numbers, has never been proved. But mathematicians are buzzing about a new attempt.
Esteemed mathematician Michael Atiyah took a crack at proving the hypothesis in a lecture at the Heidelberg Laureate Forum in Germany on September 24. Despite the stature of Atiyah — who has won the two most prestigious honors in mathematics, the Fields Medal and the Abel Prize — many researchers have expressed skepticism about the proof. So the Riemann hypothesis remains up for grabs.
Let’s break down what the Riemann hypothesis is, and what a confirmed proof — if one is ever found — would mean for mathematics.
What is the Riemann hypothesis?
The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly divisible only by 1 and themselves. Prime numbers are mysterious: They are scattered in an inscrutable pattern across the number line, making it difficult to predict where each prime number will fall ( SN Online: 4/2/08 ).
But if the Riemann zeta function meets a certain condition, Riemann realized, it would reveal secrets of the prime numbers, such as how many primes exist below a given number. That required condition is the Riemann hypothesis. It conjectures that certain zeros of the function — the points where the function’s value equals zero — all lie along a particular line when plotted ( SN: 9/27/08, p. 14 ). If the hypothesis is confirmed, it could help expose a method to the primes’ madness.
Why is it so important?
Prime numbers are mathematical VIPs: Like atoms of the periodic table, they are the building blocks for larger numbers. Primes matter for practical purposes, too, as they are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”
Haven’t people tried to prove this before?
Yep. It’s difficult to count the number of attempts, but probably hundreds of researchers have tried their hands at a proof. So far none of the proofs have stood up to scrutiny. The problem is so stubborn that it now has a bounty on its head : The Clay Mathematics Institute has offered up $1 million to anyone who can prove the Riemann hypothesis.
Why is it so difficult to prove?
The Riemann zeta function is a difficult beast to work with. Even defining it is a challenge, Ono says. Furthermore, the function has an infinite number of zeros. If any one of those zeros is not on its expected line, the Riemann hypothesis is wrong. And since there are infinite zeros, manually checking each one won’t work. Instead, a proof must show without a doubt that no zero can be an outlier. For difficult mathematical quandaries like the Riemann hypothesis, the bar for acceptance of a proof is extremely high. Verification of such a proof typically requires months or even years of double-checking by other mathematicians before either everyone is convinced, or the proof is deemed flawed.
What will it take to prove the Riemann hypothesis?
Various mathematicians have made some amount of headway toward a proof. Ono likens it to attempting to climb Mount Everest and making it to base camp. While some clever mathematician may eventually be able to finish that climb, Ono says, “there is this belief that the ultimate proof … if one ever is made, will require a different level of mathematics.”
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Riemann Hypothesis
Some equivalent statements of the Riemann Hypothesis are
Purported Disproof of the Mertens Conjecture
- Number theory
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The History and Importance of the Riemann Hypothesis
Table of Contents
Riemann Hypothesis History
The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field of number theory. It’s named after the German mathematician Bernhard Riemann, who introduced the hypothesis in 1859.
- RH: All non-trivial zeros of the Riemannian zeta function lie on the critical line.
- ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line.
- Related Article: The Extended Riemann Hypothesis and Ramanujan’s Sum: Shortest Possible Explanation
The history of the Riemann hypothesis may be considered to start with the first mention of prime numbers in the Rhind Mathematical Papyrus around 1550 BC. It certainly began with the first treatise of prime numbers in Euclid’s Elements in the 3rd century BC. It came to a – hopefully temporary – end on the 8th of August 1900 on the list of Hilbert’s famous problems. And primes are the reason why we are more than ever interested in the question of whether ERH holds or not. E.g. the RSA encryption algorithm (Rivest-Shamir-Adleman, 1977) relies on the complexity of the factorization problem FP, that it is NP-hard. FP is probably neither NP-complete nor in P but we do not know for sure. Early factorization algorithms that ran in a reasonable time had to assume the extended Riemann hypothesis (Lenstra, 1988, [1]). So what do prime numbers have in common with the Riemann hypothesis which is about a function defined as a Dirichlet series? $$ \zeta(s)=\sum_{n=1}^\infty \dfrac{1}{n^s} $$ One has to admit that what we call prime number theory today originated in the 19th century when Dirichlet began in 1837 to apply analysis to number theory. There is a large gap between Euclid and Euler who published a new proof for the infinite number of primes in 1737.
Prime Numbers
A short answer would be that $$ \zeta(s)=\sum_{n=1}^\infty \dfrac{1}{n^s}=\prod_{p\text{ prime}}\dfrac{1}{1-{p}^{-s}}. $$ This is easy to prove [5] but falls a bit short of the relationship between prime numbers and the Riemann hypothesis. E.g. the basic idea for our example of why FP can be solved quickly under the assumption of ERH is, that ERH implies the existence of relatively small primes which then can be found by fast algorithms. (See the theorems of Ankeney/Montgomery/Bach, Miller, Bach [10] and the references therein.)
Let ##1/2 \leq \theta \leq 1.## Then
$$\large{\operatorname{RH}(\theta)\, : \,\zeta(s) \text{ has no zeros in }\{\mathfrak{R}(s)>\theta\}}$$
is another generalization of the Riemann hypothesis. The original Riemann hypothesis is thus ##\operatorname{RH}(1/2)## and we know there are no zeros of the zeta-function in ##\{\mathfrak{R}(s)>1\}##. So ##\operatorname{RH}(1)## is true, but no proof is known for values of ##\theta## below. We do know ([7],[8],[9]) that $$ \large{\operatorname{RH}(\theta)\quad \Longleftrightarrow\quad \pi(x)=\operatorname{Li}(x) + O\left(x^{\theta +\varepsilon }\right)\text{ for all }\varepsilon >0} $$ where ##\operatorname{Li}(x)=\displaystyle{\int_2^x \dfrac{dt}{\log t}}## is the integral logarithm and ##\pi(x)=\left|\{p\in \mathbb{P}\,|\,p\leq x\}\right|## the prime number function (graphic from [2]). Hence ERH is connected to the question: Where are the primes?
The prime number theorem $$ \lim_{x \to \infty}\dfrac{\pi(x)}{\dfrac{x}{\log(x)}}=1 $$ was first conjectured by Gauß in 1792, however, proven by Hadamard and de la Vallée-Poussin independently a hundred years later in 1896. Their proofs were of function theoretical nature and relied on the relation of primes to the Riemannian ##\zeta##-function which was first considered by Euler in the 18th century.
Another interesting equivalent formulation of ##\operatorname{RH}(\theta)## is the following: Let ##a_{even}## be the number of integers below ##x>0## that are a product of an even number of primes, and ##a_{odd}## be the number of integers below ##x>0## that is a product of an odd number of primes, then $$ \large{\operatorname{RH}(\theta)\quad \Longleftrightarrow\quad a_{even}(x)-a_{odd}(x) = O\left(x^{\theta +\varepsilon }\right)\text{ for all }\varepsilon >0.} $$ These two equations show that the Riemann hypothesis is not only about a Dirichlet series, or the safety of some encryption algorithms. It is why I said it all began with the notion of prime numbers. We simply want to know how prime numbers are distributed. History shows that we are fascinated by prime numbers. The Riemann hypothesis ##\operatorname{RH}(1/2)## is meanwhile checked for the first ##10,000,000,000,000## zeros of the ##\zeta##-function [11], i.e. any other result than its truth would be more than surprising. In the end, we can check as many zeros as our computers can handle, it will never be a proof. However, these results above marked a huge step in the theory of prime numbers. It wasn’t long before when Euler (1707 – 1783) wrote:
“Mathematicians have hitherto strove in vain to discover any order in the sequence of prime numbers, and one is inclined to believe that this is a mystery which the human mind will never fathom. To convince oneself of this, one need only glance at the prime number tables, which some have taken the trouble to extend to 100,000, and one will at first notice that there is no order, no rule to be observed.” [12]
Early Glory
Riemann’s conjecture was only incidentally mentioned by Riemann himself, and not explicitly identified as an important problem. Riemann wrote about the zeros:
“One finds many roots within these limits, and, probably, all the roots are there. Of course, rigorous proof of this would be desirable; however, after a few unsuccessful attempts, I have left the search for it aside for the time being, since it seems unnecessary for my investigation.”
Nevertheless, he has proven that there are infinitely many roots ##s## of the ##\zeta##-function with ##\mathfrak{R}(s)=1/2## and that almost all roots are close to the critical line. Siegel has discovered these proofs in 1935 when he investigated Riemann’s estate. Riemann never published them. It was Hardy 1914 who first published a proof that there are infinitely many zeros on the critical line. A little later 1921, Hardy and Littlewood proved that there is a constant ##A>0## such that there are more than ##AT## zeros with real part ##1/2## whose (absolute) imaginary part is smaller than ##T.## It follows that there is a non-zero percentage ##B## of zeros on the critical line. Levinson showed in 1974 that ##B>1/3.##
It is not quite clear whether Hardy believed in God or was just superstitious. However, in any case, he believed God would do everything to make his life tough and complicated. One day, he was on a journey back home from a meeting with Harald Bohr (Niels Bohr’s brother) in Copenhagen. He had to take a ship and the boat he got didn’t look very trustworthy. Typically, he thought, why me? So he sent a postcard before boarding to Bohr claiming he had found the proof of Riemann’s hypothesis. When asked afterward, why, he replied: Well, if the ship sank the proof would have been lost but I would have become the most famous mathematician of my generation. God won’t allow this to happen. That way I only had to write Bohr another postcard in which I revealed to have made a mistake.
This anecdote and data demonstrate how famous the Riemann hypothesis was already at the beginning of the last century despite Riemann’s indifference to the problem that since carries his name.
Hilbert had been invited to give a lecture at the second International Congress of Mathematicians in August 1900 in Paris. He decided not to give a lecture in which he would report and appreciate what had been achieved in mathematics so far, nor to respond to Henri Poincaré’s lecture at the first International Congress of Mathematicians in 1897 on the relationship between mathematics and physics. Instead, his lecture was intended to offer a kind of programmatic outlook on future mathematics in the coming century. This objective is expressed in his introductory words:
“Who among us would not like to lift the veil that hides the future, to have a look at the forthcoming advances of our science and into the mysteries of its development during the centuries to come? What particular goals will it be that the leading mathematical minds of generations to come will aspire to? What new methods and new facts will the new centuries discover in the vast and rich field of mathematical thought?”
He, therefore, took the congress as an opportunity to compile a thematically diverse list of unsolved mathematical problems. As early as December 1899 he began to think about the subject. At the beginning of the new year, he then asked his close friends Hermann Minkowski and Adolf Hurwitz for suggestions as to which areas a corresponding lecture should cover; both read the manuscript and commented on it before the lecture. However, Hilbert only finally wrote down his list immediately before the congress – which is why it does not yet appear in the official congress program. The lecture was originally supposed to be given at the opening, but Hilbert was still working on it at the time. Now they are known as Hilbert’s 23 problems. There has been found a 24th in his estate: “How can the simplicity of a mathematical proof be measured, and how can its minimum be found?”, but the official count is 23. They are in part very specific like the first one: “Prove the continuum hypothesis.” even if not necessarily solvable, or very vague like the sixth one: “Mathematical treatment of the axioms of physics.” Here we are interested in the eighth: Prove the Riemann hypothesis, the Goldbach conjecture, and the twin prime conjecture. [3]
“Recently, significant advances have been made in the theory of the distribution of prime numbers by Hadamard, de La Vallee-Poussin, V. Mangoldt, and others. However, to completely solve the problems posed by Riemann’s treatise ‘On the Number of Primes Below a Given Size’, it is still necessary to prove the correctness of Riemann’s extremely important claim that the zeros of the function ##\zeta(s)##, which is defined by the series ##\zeta (s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\cdots ##, all have the real components ##1/2## if one disregards the well-known negative integer zeros. As soon as this proof is successful, the further task would be to examine the Riemann infinite series for the number of primes more precisely and in particular to decide whether the difference between the number of primes below a magnitude and the integral logarithm of ##x## becomes no higher than the ##\tfrac{1}{2}##th order in ##x## at infinity, and further, whether those from the first complex zeros of the function ##\zeta (s)## dependent terms of Riemann’s formula cause the local compression of the prime numbers, which one noticed when counting the prime numbers.” [13]
Yes, language was a different one a century ago. Hilbert himself classified the Riemann hypothesis as less difficult than, for example, the Fermat problem: in a lecture in 1919 he expressed the hope that a proof would be found in his lifetime, in the case of the Fermat conjecture perhaps in the lifetime of the youngest listeners; he considered the transcendence proofs in his 7th problem to be the most difficult – a problem that was solved in the 1930s by Gelfond and Theodor Schneider. The Fermat problem was solved in 1995 by Andrew Wiles and Richard Taylor as part of their proof of the modularity theorem. A proof that is not only rather long but also rather technical and complicated, so the comparison with the Riemann hypothesis is possibly not as far-fetched as it may sound.
“If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?” (David Hilbert) [6]
Let’s summarize the central limit theorem of probability theory by considering a fair coin toss and we pay +1 for heads and cash in -1 for tails. The famous gambler’s fallacy is to believe that after a long straight of heads, a tail would become more likely. This is wrong because randomness has no memory. Chances are still fifty-fifty. Even our overall gain or loss ##L(n)## after ##n## tosses isn’t zero. It is as unlikely that there will be the same number of heads as there are tails as it is that all tosses would be heads. However, it can be proven that the probability distribution of ##L(n)## converges pointwise to a normal distribution, in our case the standard normal distribution which is the statement of the central limit theorem.
We have already seen the connection between the Riemann hypothesis and randomness $$ \operatorname{RH}(\theta)\quad \Longleftrightarrow\quad a_{even}(n)-a_{odd}(n) = O\left(n^{\theta +\varepsilon }\right)\text{ for all }\varepsilon >0. $$ Let us consider the Liouville function ##\lambda (n)=(-1)^{\#\text{ prime factors of }n}## and remember that ##a_{even/odd}(n)## counted the number of integers below ##n## that are a product of an even/odd number of primes. Then $$ L(n)=\sum_{k=1}^n \lambda (k)= a_{even}-a_{odd} $$ and the central limit theorem says $$ \lim_{n \to \infty}\dfrac{L(n)}{n^{\varepsilon +1/2}}=0 \text{ for all }\varepsilon >0 \Leftrightarrow L(n)=O(n^{\varepsilon +1/2}) \Leftrightarrow RH(1/2)$$ This means that the pseudo-randomness of the distribution of prime numbers is almost independent and identical, i.e. truly random. It seems Euler was right once more.
Number Theory
We already mentioned the fascination with prime numbers and their central meaning in number theory. It is no coincidence that there are three famous problems listed under Hilbert’s 8th problem:
- Riemann Hypothesis ##a_{even}(n)-a_{odd}(n)=O(n^{\varepsilon +1/2})##
- Goldbach’s Conjecture Every even integer greater than ##2## is the sum of two primes.
- Twin Prime Conjecture There are infinitely many pairs ##(p,p+2)## of prime numbers ##p.##
Neither of these conjectures are proven although they have been tested for incredibly large amounts of numbers computationally. They all have to do with prime numbers. An integer ##p## is prime, if and only if ## (p-1)!\equiv -1 {\pmod p}## ( Wilson’s theorem ), and a pair ##(p,p+2)## is a pair of primes, if and only if ##4\cdot ((p-1)!+1)+p \equiv 0 {\pmod {p\cdot (p+2)}}## ( Clement’s theorem ).
Goldbach’s conjecture or the strong Goldbach conjecture has a weaker version: Every odd number greater than ##5## is the sum of three primes. Since ##3## is prime and ##2n+1=2n-2+3=p+q+3,## the strong version implies the weaker, which has been partially solved. On one hand, is it true in case the extended Riemann hypothesis holds, and on the other hand, it holds for sufficiently large numbers? If ##R(n)## is the number of representations of ##n## as the sum of three prime numbers, then ( Vinogradov’s theorem ) $$ R(n)=\dfrac{n^2}{2(\log n)^3}\underbrace{\left(\prod_{p\mid n}\left(1-\dfrac{1}{(p-1)^2}\right)\right)\left(\prod_{p\nmid n}\left(1+\dfrac{1}{(p-1)^3}\right)\right)}_{=:G(n)}+O\left(\dfrac{n^2}{(\log n)^4}\right) $$ and it can be shown that ##G(2n)=0,## ##G(2n-1)\geq 1,## and ##G(2n-1)## is asymptotically of order ##O(1),## hence ##R(2n-1)>0## for sufficiently large ##n.##
There have been quite a few attempts to tackle the problem by physical methods, especially lately. This is quite surprising since mathematics is a deductive science and physics a descriptive science. One can check the results of theoretical models in the physical world, but how should real-world observations contribute to a mathematical conjecture? The origin of such a connection, however, isn’t quite new. David Hilbert and Pólya György had already noticed that the Riemann hypothesis would follow if the zeros were eigenvalues of an operator ##({\tfrac{1}{ 2}}+iT)## where ##T## is a Hermitian (i.e. self-adjoint) operator, which therefore has only real eigenvalues, similar to the Hamiltonian operators in quantum mechanics. Further considerations in this direction end up in the theory of quantum chaos. Other connections have been drawn to statistical mechanics [3], or one-dimensional quasi-crystals [15]. We even had a visitor on Physics Forums who attempted to access it via the hydrogen atom. All these thoughts are based on parallels between the Riemann hypothesis and probability distributions and are often due to similarities in formulas.
In addition to numerous applications in many areas of mathematics, the Riemann Hypothesis is also of interest in cryptology. For example, the RSA cryptosystem uses large prime numbers to construct both public and private keys. Its security is based on the fact that conventional computers do not yet have an efficient algorithm for dividing a number into its prime factors, i.e. to solve FP. The theory behind RSA requires only results from elementary number theory. In 1976, again based on simple number theory and using Fermat’s little theorem, Miller developed a deterministic primality test that works assuming the extended Riemann Hypothesis [16]. In 1980, Michael O. Rabin used Miller’s results to develop a probabilistic test that worked independently of the extended Riemann hypothesis [17]. Through the work of Bach in 1990, this so-called Miller-Rabin test can be converted into a deterministic test that runs with the speed ##O(\log(n)^{2})##, again assuming the extended Riemann Hypothesis [10]. All the connections between the Riemann hypothesis and cryptology are at their core due to their meaning for the distribution of prime numbers.
[1] A.K. Lenstra, Fast and rigorous factorization under the generalized Riemann hypothesis, Indagationes Mathematicae (Proceedings), Volume 91, Issue 4, 1988, Pages 443-454, ISSN 1385-7258
[2] German Wikipedia, Primzahlsatz https://commons.wikimedia.org/wiki/File:PrimeNumberTheorem.svg https://de.wikipedia.org/wiki/Riemannsche_Vermutung
[3] Wikipedia https://de.wikipedia.org/wiki/Riemannsche_Vermutung https://de.wikipedia.org/wiki/Goldbachsche_Vermutung https://de.wikipedia.org/wiki/Hilbertsche_Probleme https://en.wikipedia.org/wiki/Hilbert%27s_problems https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture
[4] Otto Forster, München 2017/2018, 8. Äquivalenzen zur Riemannschen Vermutung https://www.mathematik.uni-muenchen.de/~forster/v/zrh/vorlzrh_chap8.pdf
[5] The Extended Riemann Hypothesis and Ramanujan’s Sum https://www.physicsforums.com/insights/the-extended-riemann-hypothesis-and-ramanujans-sum/
[6] AZ Quotes https://www.azquotes.com/author/6689-David_Hilbert
[7] Maier, Haase, Analytical Number Theory, Ulm 2007 (in German) https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.zawa/lehre/12sem-pz/Analytische_Zahlentheorie_SS_2007.pdf
[8] W. Dittrich, On Riemann’s Paper, “On the Number of Primes Less Than a Given Magnitude”, Tübingen 2017 https://arxiv.org/pdf/1609.02301.pdf
[9] Bernhard Riemann, On the Number of Prime Numbers less than a Given Quantity. (Über die Anzahl der Primzahlen unter einer gegebenen Grösse. [Monatsberichte der Berliner Akademie, November 1859.]) Translated by David R. Wilkins, 1998 https://www.claymath.org/sites/default/files/ezeta.pdf
[10] Eric Bach, Explicit Bounds for Primality Testing and Related Problems, Mathematics of Computation, Volume 55, Number 191, July 1990, pages 355-380 https://www.ams.org/journals/mcom/1990-55-191/S0025-5718-1990-1023756-8/S0025-5718-1990-1023756-8.pdf
[11] X. Gourdon, The 10E13 first zeros of the Riemann Zeta function, and zeros computation at very large height (2004) http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf
[12] Jean Dieudonné, Geschichte der Mathematik 1700-1900, Vieweg Verlag 1985
[13] Julian Havil, Gamma. Springer-Verlag, Berlin et al. 2007, p. 244-245.
[14] H.M.Edwards, Riemann’s Zeta Function, Dover Publications Inc., 2003 (315 pages) https://www.amazon.de/Riemanns-Function-Dover-Mathematics-Applied/dp/0486417409/ref=asc_df_0486417409/
[15] Freman Dyson, Birds and Frogs, 2009 https://www.ams.org//notices/200902/rtx090200212p.pdf
[16] Gary L. Miller, Riemann’s Hypothesis and Tests for Primality, Journal of Computer and System Sciences, 1976, 13(3), p. 300–317.
[17] M. O. Rabin, Probabilistic algorithm for testing primality, Journal of Number Theory, 1980, 12(1), p. 128–138.
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- Riemann Hypothesis
- 1.1 Critical Strip
- 1.2 Critical Line
- 2.1 Trivial Zeroes of Riemann Zeta Function are Even Negative Integers
- 2.2 First zero
- 3 Hilbert $23$
- 4 Also known as
- 6 Source of Name
- 7 Historical Note
All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.
Critical Strip
Let $s = \sigma + i t$.
The region defined by the equation $0 < \sigma < 1$ is known as the critical strip .
Critical Line
The line defined by the equation $\sigma = \dfrac 1 2$ is known as the critical line .
Hence the popular form of the statement of the Riemann Hypothesis :
Some of the zeroes of Riemann $\zeta$ function are positioned as follows:
Trivial Zeroes of Riemann Zeta Function are Even Negative Integers
Let $\rho = \sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip :
These are called the trivial zeros of $\zeta$.
The first zero of the Riemann $\zeta$ function is positioned at:
Hilbert $23$
This problem is no. $8a$ in the Hilbert $23$ .
Also known as
The Riemann hypothesis is also known as the zeta hypothesis .
- All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip
- Critical Line Theorem : an infinite number of nontrivial zeroes exist on the critical line , whatever their multiplicity
Source of Name
This entry was named for Georg Friedrich Bernhard Riemann .
Historical Note
The Riemann Hypothesis was stated by Bernhard Riemann in his $1859$ article Ueber die Anzahl der Primzahlen under einer gegebenen Grösse .
It is the last remaining statement which has not been resolved is the Riemann Hypothesis .
This problem is the first part of no. $8$ in the Hilbert 23 , and also one of the Millennium Problems , the only one to be in both lists.
In Riemann 's words, in his posthumous papers:
As Jacques Salomon Hadamard put it:
In $1914$, Godfrey Harold Hardy proved the Critical Line Theorem , that there exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line .
This was again demonstrated in $1921$, by Godfrey Harold Hardy together with John Edensor Littlewood .
In $1974$, Norman Levinson demonstrated that At Least One Third of Zeros of Riemann Zeta Function on Critical Line .
By $1983$, systematic exploration of the critical strip with the aid of computers had shown that the first $3 \, 500 \, 000$ nontrivial zeroes were all located on the critical line .
While this is compelling, it is far from being a proof .
In December $1984$, it was announced that Hideya Matsumoto had found a proof, but this was shown to be flawed.
- 1983: François Le Lionnais and Jean Brette : Les Nombres Remarquables ... (previous) ... (next) : $0,5$
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The Riemann hypothesis, a Step Closer to Being Solved
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The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved Scientific American Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed … link
The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.
But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.
Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”
The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.
Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.
Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.
There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.
Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 BCE that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.
As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line….
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What is the Riemann-Zeta function?
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
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- 33 $\begingroup$ A good example of a question that is asked by someone genuinely interested in math but is looking for an accessible way into more advanced number theory that he would otherwise have no other means of finding. $\endgroup$ – Justin L. Commented Jul 23, 2010 at 6:47
5 Answers 5
Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you're doing number theory then you want to think of the prime numbers as acting "independently": knowing that a number is divisible by $p$ should give you no information about whether it's divisible by $q$.
That quickly leads you to the following realization: you should choose the exponent of each prime in the prime factorization independently. So how should you choose these? It turns out that the probability distribution on the non-negative integers with maximum entropy and a given mean is a geometric distribution, as explained for example by Keith Conrad here . So let's take the probability that the exponent of $p$ is $k$ to be equal to $(1 - r_p) r_p^k$ for some constant $r_p$.
This gives the probability that a positive integer $n = p_1^{e_1} ... p_k^{e_k}$ occurs as
$\displaystyle C \prod_{i=1}^{k} r_p^{e_i}$
where $C = \prod_p (1 - r_p)$. So we need to choose $r_p$ such that this product converges. Now, we'd like the probability that $n$ occurs to be monotonically decreasing as a function of $n$. It turns out (and this is a nice exercise) that this is true if and only if $r_p = p^{-s}$ for some $s > 1$ (since $C$ has to converge), which gives the probability that $n$ occurs as
$\frac{ \frac{1}{n^s} }{ \zeta(s)}$
where $\zeta(s)$ is the zeta function.
One way of thinking about this argument is that $\zeta(s)$ is the partition function of a statistical-mechanical system called the Riemann gas . As $s$ gets closer to $1$, the temperature of this system increases until it would require infinite energy to make $s$ equal to $1$. But this limit is extremely important to understand: it is the limit in which the probability distribution above gets closer and closer to uniform. So it's not surprising that you can deduce statistical information about the primes by studying the behavior as $s \to 1$ of this distribution.
Let me mention two other reasons to care about the limit as $s \to 1$ of the above distribution. First, the basic reason to think of the primes as acting independently is the Chinese Remainder Theorem. Second, a natural reason to look at a distribution where the probability that a number has exactly $k$ factors of $p$ is $(1 - p^{-1}) p^{-k}$ is that this is precisely the distribution you get on the residues $\bmod p^n$ for $k < n$. In fact, I believe this can be upgraded to the corresponding statement about Haar measure on the $p$-adic integers.
- 35 $\begingroup$ I feel like I've just been let in on a mathematical secret. $\endgroup$ – I. J. Kennedy Commented Oct 27, 2010 at 4:48
Giving an explanation in layman's terms is always going to be challenging, given that the Riemann-Zeta function (and related hypothesis) inevitably lies in the domain of abstract mathematics, but I shall do my best.
The Riemann-Zeta function is a complex function that tells us many things about the theory of numbers. Its mystery is increased by the fact it has no closed form - i.e. it can't be expressed a single formula that contains other standard (elementary) functions.
Although there are many different ways of expressing the Riemann-Zeta function (the Wikipedia article gives several), it can ultimately be derived from the following simple series of real numbers:
$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s},\quad\Re(s)\gt1$
by extending it into the complex plane .
The reason this strange and esoteric function is so famous and actively discussed in mathematics is due to the Riemann hypothesis - proposed in 1859 by the great Bernhard Riemann and still unsolved. The Wiki article states the problem in quite simple terms:
The Riemann zeta-function ζ(s) is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, −4, −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Riemann zeta function is 1/2. Thus the non-trivial zeros should lie on the critical line, 1/2 + it, where t is a real number and i is the imaginary unit.
Although the conjecture (it is only that at the moment) has many consequences for mathematics (number theory in particular), the primary one, at least the one Riemann originally proposed, is about the distribution of prime numbers . In other words, it tells us with great precision what the average gaps between primes are as we move to greater and greater numbers. Many of the other implications are rather more esoteric, though perhaps equally important for pure mathematicians.
- 2 $\begingroup$ Why does the zeta function have zeros at negative even integers? Wouldn't it simplify to a divergent sum of powers of natural numbers? $\endgroup$ – JacksonFitzsimmons Commented Jul 16, 2015 at 18:02
- $\begingroup$ @JacksonFitzsimmons The important part about $\zeta(s)=\sum_{n=1}^\infty\frac1{n^s}$ is the $\Re(s)>1$. To evaluate when $\Re(s)\le1$, use analytic continuation. $\endgroup$ – Simply Beautiful Art Commented Dec 21, 2016 at 0:50
The above answers give excellent explanations about why the zeta function has close connections to number theory, but I thought I'd mention something about why the Riemann Hypothesis should matter so much.
By taking the logarithm and then differentiating the zeta function, one gets the formula $$\frac{\zeta'(s)}{\zeta(s)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s}$$
for $\Re(s)>1$, where $\Lambda(n)$ is the von Mangoldt function which takes the value $\log p$ at powers of primes $p$, and is 0 everywhere else. Think of it as a weighted way of counting the primes (the prime number theorem tells us that $\log p$ is the natural weight to choose).
Much of analytic number theory proceeds by choosing a weight of the set we wish to consider (often the primes), and then encoding this weighting in a so-called Dirichlet series (an infinite sum of the form above). We can then use analysis to study this series and get lots of useful information.
In this case, then, the function we need to study to get information about the primes is $\frac{\zeta'(s)}{\zeta(s)}$, which we can study using complex analysis.
In complex analysis, a good slogan is 'the only things that matter are zeros and poles' (effectively points where the function shoots off to infinity).
Hence to understand the prime numbers, we just need to understand the zeros and poles of $\frac{\zeta'(s)}{\zeta(s)}$ - we know about the simple pole at $s=1$, we know there aren't any other zeros where it counts, and we also know that the only other poles are at zeros of $\zeta(s)$ (roughly because dividing by zero causes infinity).
In other words, if we knew where these zeros are (i.e. the Riemann hypothesis) we can work with $\frac{\zeta'(s)}{\zeta(s)}$ in all kinds of clever ways to get good results on the prime numbers.
More specifically, in the usual contour proof of the prime number theorem, knowing that there aren't any other zeros in $\Re(s)>1/2$ would allow us to shift the contour further to the left, reducing the error term in the result to (roughly) $O(\sqrt{x})$.
Here there is another attempt at an explanation.
We know that the sum of the inverse of the positive numbers, $1 + 1/2 + 1/3 + \cdots$, diverges. Euler shown that the sum of the inverse of the squares, $1/(1^2) + 1/(2^2) + 1/(3^2) + \cdots$, has a finite sum, namely $\pi^2/6$. Mathematicians love to generalize things, so they thought at the function
$\displaystyle f(x)=\sum_{n=1}^\infty\dfrac1{n^x}$
which is defined for $x \gt 1$. But this was not enough: they decided that the variable could be a complex number and not a real one. There is a standard tecnique ( Analytic continuation ) which allows us to extend the function to nearly all the complex plane. So we now have a function which formally is
$\displaystyle \zeta(s)=\sum_{n=1}^\infty\dfrac1{n^s}$
(the variable being $s$ and not $x$ to show that we are dealing with complex numbers) but is not computed in this way. Just to make an example, $\zeta(0)=-1/2$, and sum of an infinity of ones is not $-1/2$. :-)
It may be shown that for $s = -2n$ ($n$ positive integer) $\zeta(s) = 0$. But there are infinite other point $s'=(x,y)$ where $\zeta(s') = 0$. For all of these points, $0 \lt x \lt 1$; Riemann's hypothesis says that for all such points $x = 1/2$. If it were true, we could have the best asymptotic expression to count $\pi(n)$, that is the number of primes below $n$.
Why does the function pop up when we talk about primes? I don't know, but in the case of integer values Euler proved that
$\displaystyle\sum_{n=1}^\infty\frac1{n^s}=\prod_{p \text{ prime}}\frac1{1-p^{-s}}$
Maybe this could be a good start.
- $\begingroup$ The product formula is valid for all $s$ with $\Re(s)>1$; it's of great importance as to why the zeta-function (and its cousins the L-functions) relates to the distribution of prime numbers. $\endgroup$ – Akhil Mathew Commented Jul 23, 2010 at 11:45
- 1 $\begingroup$ Correction: zeta(0) is -1/2, not 1/2. Also, I think it is misleading to say analytic continuation is a "standard technique" to extend the zeta-function. Whether a function has an analytic continuation to some larger region is a property, but checking where that property works often depends on special aspects of the particular function under consideration. $\endgroup$ – KCd Commented Feb 3, 2011 at 3:45
- $\begingroup$ @KCd =D Fixed the annoying little error. $\endgroup$ – Simply Beautiful Art Commented Dec 21, 2016 at 0:53
The key point is that the Riemann zeta function is a function whose properties encode properties about the prime numbers. As mentioned by Noldorin, in order to fully understand the Riemann zeta function you need to "analytically continue it to the complex plane" which is a tricky process which takes serious study. Fortunately for some easier properties of the primes you can just use the definition of the zeta function for real s.
Claim (due to Euler): The fact that $\zeta(s)$ goes to infinity as $s\to 1$ tells you that there are infinitely many primes.
Sketch of proof: Use the "Euler factorization" mentioned by mau (expand the RHS as a geometric series and then multiply it out using unique factorization into primes):
$\displaystyle\sum_{n=1}^\infty \frac{1} {n^s} = \prod_{p\in\text{prime}} \frac{1} {1-p^-s}$
Now take log of both sides to get: $\displaystyle\log \zeta(s) = \sum_{p\in\text{prime}} \log \frac{1} {1-p^-s}.$
Now use the taylor series for \log and send s to one. You'll get that the left hand side goes to infinity ( Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? ), while the right hand side looks like $\sum 1/p$ + bounded terms. So there must be infinitely many primes.
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Riemann Zeta Function
which is sometimes known as a p -series . The Riemann zeta function can also be defined in terms of multiple integrals by and as a Mellin transform by It appears in the unit square integral The Riemann zeta function satisfies the reflection functional equation Hasse (1930) also proved the related globally (but more slowly) convergent series The Riemann zeta function can also be defined in the complex plane by the contour integral The Riemann zeta function can be split up into (Spanier and Oldham 1987). (Lehman 1960, Hardy and Wright 1979). Furthermore, giving the first few as (OEIS A114875 ). The identity which is known as the Euler product formula (Hardy 1999, p. 18; Krantz 1999, p. 159), and called "the golden key" by Derbyshire (2004, pp. 104-106). The formula can also be written (Havil 2003, p. 209). An integral for positive even integers is given by and integrals for positive odd integers are given by which gives This value is related to a deep result in renormalization theory (Elizalde et al. 1994, 1995, Bloch 1996, Lepowski 1999). It is apparently not known if the value (OEIS A059750 ) can be expressed in terms of known mathematical constants. This constant appears, for example, in Knuth's series . (Cohen 2000). the first few values can then be written (Plouffe 1998). Another set of related formulas are (Plouffe 2006). G. Huvent (2002) found the beautiful formula Sums involving integers multiples of the argument include Other unexpected sums are (Tyler and Chernhoff 1985; Boros and Moll 2004, p. 248) and ( 125 ) is a special case of Considering the sum (B. Cloitre, pers. comm., Dec. 11, 2005; cf. Borwein et al. 2000, eqn. 27).
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Referenced on Wolfram|AlphaCite this as:. Sondow, Jonathan and Weisstein, Eric W. "Riemann Zeta Function." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannZetaFunction.html Subject classifications
Note that the real part of these zeroes is always 1/2. Riemann’s famous hypothesis is that all nontrivial zeroes of the zeta function lie along the line ${\rm Re}(z) = 1/2$. The Riemann hypothesis is widely regarded as the most significant outstanding unsolved problem in mathematics. For instance, the Clay Mathematics Institute lists the Riemann hypothesis as one of its “millennium problems,” the solution to which would qualify for an award of one million U.S. dollars. For some additional background on the Riemann hypothesis, see this article written by Peter Sarnak for the Clay Institute. Consequences of the Riemann hypothesisIf the Riemann hypothesis is true, many important results would hold. Here are just two:
For details and some other examples, see this Wikipedia article . The Griffin-Ono-Rolen-Zagier resultIn a remarkable new paper , published in the Proceedings of the National Academy Of Sciences , the four mathematicians have resurrected a line of reasoning, long thought to be dead, originally developed by Johan Jensen and George Polya . The proof relies on “Jensen polynomials,” which for an arbitrary real sequence $(\alpha(0), \alpha(1), \alpha(2), \ldots)$, integer degree $d$ and shift $n$ are defined as: $$J_\alpha^{d,n} (x) = \sum_{j=0}^d {d \choose j} \alpha(n+j) x^j.$$ We say that a polynomial with real coefficients is hyperbolic if all of its zeroes are real. Define $\Lambda( s) = \pi^{-s/2} \Gamma(s/2) \zeta(s) = \Lambda(1-s)$. Consider now the sequence of Taylor coefficients $(\gamma(n), n \geq 1)$ defined implicitly by $$(4z^2 – 1) \Lambda(z+1/2) \; = \; \sum_{n=0}^\infty \frac{\gamma(n) z^{2n}}{n!}.$$ Polya proved that the Riemann hypothesis is equivalent to the assertion that the Jensen polynomials associated with the sequence $(\gamma(n))$ are hyperbolic for all nonnegative integers $d$ and $n$. What Griffin, Ono, Rolen and Zagier have shown is that for $d \geq 1$, the associated Jensen polynomials $J_\gamma^{d,n}$ are hyperbolic for all sufficiently large $n$. This is not the same as for every $n$, but it certainly is a remarkable advance. In addition, the four authors proved that for $1 \leq d \leq 8$, that the associated Jensen polynomials are indeed hyperbolic for all $n \geq 0$. Previous to this result, the best result was for $1 \leq d \leq 3$ and all $n \geq 0$. Ken Ono emphasizes that he and the other authors did not invent any new techniques or new mathematical objects. Instead, the advantage of their proof is its simplicity (the paper is only eight pages long!). The idea for the paper was a “toy problem” that Ono presented for entertainment to Zagier during a recent conference celebrating Zagier’s 65th birthday. Ono thought that the problem was essentially intractable and did not expect Zagier to make much headway with it, but Zagier was enthused by the challenge and soon had sketched a solution. Together with the other authors, they fleshed out the solution and then extended it to a more general theory. Kannan Soundararajan, a Stanford mathematician who has studied the Riemann Hypothesis, said “The result established here may be viewed as offering further evidence toward the Riemann Hypothesis, and in any case, it is a beautiful stand-alone theorem.” How much longer?The authors emphasize that their work definitely falls short of a full proof of the Riemann hypothesis. For all they know, the hypothesis may still turn out to be false, or that what remains in this or any other proposed proof outline is so difficult that it may defy efforts to prove for many years to come. But the result is definitely encouraging. It should be mentioned that some other manuscripts have circulated with authors claiming proofs, at least a few of which are by mathematicians with very solid credentials. However, none of these has ever gained any traction, so the only safe conclusion is that the Riemann hypothesis remains unproven and may be as difficult as ever. Will it still be unproven 100 years from now? Stay tuned (if any of us are still around in 2119). Comments are closed. Recent Posts
Academic DepartmentsHelpful resources, support science. Please contact Elizabeth Chadis if you are considering a gift to the School of Science. The Riemann Hypothesis, the Biggest Problem in Mathematics, Is a Step Closer to Being SolvedRead it at Scientific American
Riemann Hypothesis/Biography of Riemann Bernhard Riemann was born in the Kingdom of Hanover, in modern Germany, in 1826. From an early age, Riemann directed much of his interest towards theology and philology. While he excelled in mathematics, he was rather timid and shy, and neglected to display much of his ability. When Riemann attended the University of Göttingen, he initially aspired to study Theology. At university, Riemann met Gauss, who advised him to give up his theological studies, and pursue mathematics. He later transferred to the University of Berlin to read mathematics, where several notable mathematicians, including Steiner, Jacobi and Dirichlet (from whom he would borrow concepts for his later studies). Riemann died from tuberculosis on a trip to Italy in 1866. Despite his early death, Riemann left a significant legacy that can still be seen in mathematics to this day. Riemann's academic work primarily concerned analysis, number theory and differential geometry. He is credited with his contribution of the Riemann integral, the first formal, rigorous, definition of an integral, Riemann surfaces and Riemannian geometry, the latter being later used by Einstein as part of his theory of General Relativity. This piece concerns Riemann's research in real analysis, specifically prime numbers, where he explored the distribution of prime numbers, later proposing the Riemann Hypothesis in 1859. 
Navigation menuRiemann HypothesisIn mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.  The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first nontrivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. Animation showing in 3D the Riemann zeta function critical strip (blue), critical line (red) and zeroes (cross between red and orange): [x,y,z] = [Re(ζ(r + it), Im(ζ(r + it), t] with 0.1 ≤ r ≤ 0.9 and 1 ≤ t ≤ 51  Riemann zeta function along critical line Re(s) = 1/2 (real values are on the horizontal axis and imaginary values are on the vertical axis): Re(ζ(1/2 + it), Im(ζ(1/2 + it) with t ranging between −30 and 30 The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit . More Information Found On [ ]
 www.springer.com The European Mathematical Society
Riemann hypothesesin analytic number theory Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the zeta-function \begin{equation} \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it, \end{equation} and the expression via these zeros of the number of prime numbers not exceeding a real number $x$. One of the Riemann hypotheses has neither been proved nor disproved: All non-trivial zeros of the zeta-function $\zeta(s)$ lie on the straight line $\operatorname{Re} s = 1/2$. For the list of all 5 conjectures see Zeta-function .
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This plot of Riemann's zeta (ζ) function (here with argument z) shows trivial zeros where ζ(z) = 0, a pole where ζ(z) = , the critical line of nontrivial zeros with Re() = 1/2 and slopes of absolute values.In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 / 2 .
The Riemann hypothesis is a mathematical question ( conjecture ). Finding a proof of the hypothesis is one of the hardest and most important unsolved problems of pure mathematics. [1] Pure mathematics is a type of mathematics that is about thinking about mathematics. This is different from trying to put mathematics into the real world.
Local zeta function. In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse-Weil zeta function) is defined as. where V is a non-singular n -dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension ...
First published in Riemann's groundbreaking 1859 paper (Riemann 1859), the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of s other than -2, -4, -6, ... such that zeta(s)=0 (where zeta(s) is the Riemann zeta function) all lie on the "critical line" sigma=R[s]=1/2 (where R[s] denotes the real part of s). A ...
A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography.. The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics.It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert ...
The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re (s) = σ > 1, the function can be written as a converging summation or as an integral: where. is the gamma function.
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L -functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the ...
The Riemann hypothesis, posited in 1859 by German mathematician Bernhard Riemann, is one of the biggest unsolved puzzles in mathematics. The hypothesis, which could unlock the mysteries of prime ...
The Riemann Hypothesis is a famous conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta function have real part.From the functional equation for the zeta function, it is easy to see that when .These are called the trivial zeros. This hypothesis is one of the seven millenium questions.. The Riemann Hypothesis is an important problem in the study of ...
The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field of number theory. It's named after the German mathematician Bernhard Riemann, who introduced the hypothesis in 1859. RH: All non-trivial zeros of the Riemannian zeta function lie on the critical line.
It is the last remaining statement which has not been resolved is the Riemann Hypothesis . This problem is the first part of no. 8 8 in the Hilbert 23, and also one of the Millennium Problems, the only one to be in both lists. In Riemann 's words, in his posthumous papers: [ These theorems ] follow from an expression for the function ζ(s) ζ ...
The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on. Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to ...
Bernhard Riemann. Georg Friedrich Bernhard Riemann ( German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] ⓘ; [ 1][ 2] 17 September 1826 - 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
Riemann Hypothesis/The hypothesis. < Riemann Hypothesis. Theorem 1. Proof. Consider the functional equation for Zeta, Notice that for , the sine term evaluates to which evaluates to 0 for all integers , hence for all natural . Definition 1. These zeroes are referred to as trivial zeroes. As a set,
The Riemann zeta function has two kinds of zeros, trivial zeroes (at the negative even integers, −2 − 2, −4 − 4, et c.) and the non-trivial zeroes. There are infinitely many non-trivial zeroes and all of them are known to lie in the strip having real parts between 0 0 and 1 1 (in detail, the strip {x +iy ∈C ∣ 0 < x < 1} { x + i y ...
The reason this strange and esoteric function is so famous and actively discussed in mathematics is due to the Riemann hypothesis - proposed in 1859 by the great Bernhard Riemann and still unsolved. The Wiki article states the problem in quite simple terms: The Riemann zeta-function ζ(s) is defined for all complex numbers s ≠ 1. It has zeros ...
The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved ...
In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis.It states that the nontrivial zeros of all automorphic L-functions lie on the critical line + with a real number variable and the imaginary unit.. The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L-functions lie on the ...
Riemann Hypothesis. The Riemann hypothesis is considered to be one of the most important conjectures within pure mathematics, which has stood unsolved for over 150 years. This wikibook seeks to explore the hypothesis, its history, and its current status.
The Riemann hypothesis was first posed by the German mathematician Georg Friedrich Bernhard Riemann in 1859, in a paper where he observed that questions regarding the distribution of prime numbers were closely tied to a conjecture regarding the behavior of the "zeta function," namely the beguilingly simple expression ζ(s) = ∞ ∑ n=1 1 ...
The Riemann Hypothesis, the Biggest Problem in Mathematics, Is a Step Closer to Being Solved. Read it at Scientific American. Published July 01, 2024 Tagged Larry Guth Mathematics. MIT School of Science. Biology; Brain and Cognitive Sciences; Chemistry; Earth, Atmospheric and Planetary Sciences;
Riemann Hypothesis/Biography of Riemann. Bernhard Riemann was born in the Kingdom of Hanover, in modern Germany, in 1826. From an early age, Riemann directed much of his interest towards theology and philology. While he excelled in mathematics, he was rather timid and shy, and neglected to display much of his ability.
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard ...
Riemann hypotheses. in analytic number theory. Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the zeta-function \begin {equation} \zeta (s)=\sum_ {n=1}^\infty\frac {1} {n^s},\quad s=\sigma+it, \end {equation} and the expression via these zeros of the number of prime numbers not ...