The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.
Definitionedit
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:
is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1.
Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to $\operatorname {Re} (s)>1.$^{[4]}
The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1, the series is the harmonic series which diverges to +∞, and
where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):
Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes Π_{p}p/p − 1) implies that there are infinitely many primes.^{[5]} Since the logarithm of p/p − 1 is approximately 1/p, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.
The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) p is 1/p. Hence the probability that s numbers are all divisible by this prime is 1/p^{s}, and the probability that at least one of them is not is 1 − 1/p^{s}. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/nm). Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.
Proof of Riemann's functional equation
A proof of the functional equation proceeds as follows:
We observe that if $\sigma >0$, then
which is the functional equation.
E. C. Titchmarsh (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford: Oxford Science Publications. pp. 21–22. ISBN 0-19-853369-1. Attributed to Bernhard Riemann.
The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (the alternating zeta function):
where the η-series is convergent (albeit non-absolutely) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine^{[6]}^{[7]}).
Riemann also found a symmetric version of the functional equation applying to the xi-function:
The $\pi ^{-s/2}\Gamma (s/2)$ factor was not well-understood at the time of Riemann, until John Tate's (1950) thesis, in which it was shown that this so-called "Gamma factor" is in fact the local L-factor corresponding to the Archimedean place, the other factors in the Euler product expansion being the local L-factors of the non-Archimedean places.
Zeros, the critical line, and the Riemann hypothesisedit
The functional equation shows that the Riemann zeta function has zeros at −2, −4,.... These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin πs/2 being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip $\{s\in \mathbb {C} :0<\operatorname {Re} (s)<1\}$, which is called the critical strip. The set $\{s\in \mathbb {C} :\operatorname {Re} (s)=1/2\}$ is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.^{[8]}
For the Riemann zeta function on the critical line, see Z-function.
First few nontrivial zeros^{[9]}^{[10]}
Zero
1/2 ± 14.134725 i
1/2 ± 21.022040 i
1/2 ± 25.010858 i
1/2 ± 30.424876 i
1/2 ± 32.935062 i
1/2 ± 37.586178 i
1/2 ± 40.918719 i
Number of zeros in the critical stripedit
Let $N(T)$ be the number of zeros of $\zeta (s)$ in the critical strip $0<\operatorname {Re} (s)<1$, whose imaginary parts are in the interval $0<\operatorname {Im} (s)<T$.
Trudgian proved that, if $T>e$, then^{[11]}
In 1914, Godfrey Harold Hardy proved that ζ (1/2 + it) has infinitely many real zeros.^{[12]}
Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of ζ (1/2 + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N_{0}(T) the total number of zeros of odd order of the function ζ (1/2 + it) lying in the interval (0, T].
For any ε > 0, there exists a T_{0}(ε) > 0 such that when
$T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{4}}+\varepsilon },$
the interval (T, T + H] contains a zero of odd order.
For any ε > 0, there exists a T_{0}(ε) > 0 and c_{ε} > 0 such that the inequality
$N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon }H$
holds when
$T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{2}}+\varepsilon }.$
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
Zero-free regionedit
The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.^{[13]} A better result^{[14]} that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever $\sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}$ and |t| ≥ 3.
In 2015, Mossinghoff and Trudgian proved^{[15]} that zeta has no zeros in the region
$\sigma \geq 1-{\frac {1}{5.573412\log |t|}}$
for |t| ≥ 2.
This is the largest known zero-free region in the critical strip for $3.06\cdot 10^{10}<|t|<\exp(10151.5)\approx 5.5\cdot 10^{4408}$.
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.
Other resultsedit
It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γ_{n}) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)
In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + 14.13472514...i (OEIS: A058303). The fact that
$\zeta (s)={\overline {\zeta ({\overline {s}})}}$
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = 1/2.
It is also known that no zeros lie on the line with real part 1.
where B_{2n} is the 2n-th Bernoulli number.
For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see Special values of L-functions.
For nonpositive integers, one has
$\zeta (-n)=-{\frac {B_{n+1}}{n+1}}$
for n ≥ 0 (using the convention that B_{1} = 1/2).
In particular, ζ vanishes at the negative even integers because B_{m} = 0 for all odd m other than 1. These are the so-called "trivial zeros" of the zeta function.
This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts (Ramanujan summation) such as string theory.^{[16]} Analogously, the particular value
$\zeta (0)=-{\tfrac {1}{2}}$
can be viewed as assigning a finite result to the divergent series 1 + 1 + 1 + 1 + ⋯.
is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?^{[20]}
The value
for every complex number s with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.
The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.
Universalityedit
The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.^{[21]} More recent work has included effective versions of Voronin's theorem^{[22]} and extending it to Dirichlet L-functions.^{[23]}^{[24]}
Estimates of the maximum of the modulus of the zeta functionedit
Let the functions F(T;H) and G(s_{0};Δ) be defined by the equalities
Here T is a sufficiently large positive number, 0 < H ≪ log log T, s_{0} = σ_{0} + iT, 1/2 ≤ σ_{0} ≤ 1, 0 < Δ < 1/3. Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ Re(s) ≤ 1.
The case H ≫ log log T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.
Anatolii Karatsuba proved,^{[25]}^{[26]} in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates
is called the argument of the Riemann zeta function. Here arg ζ(1/2 + it) is the increment of an arbitrary continuous branch of arg ζ(s) along the broken line joining the points 2, 2 + it and 1/2 + it.
There are some theorems on properties of the function S(t). Among those results^{[27]}^{[28]} are the mean value theorems for S(t) and its first integral
$S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u$
on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for
$H\geq T^{{\frac {27}{82}}+\varepsilon }$
contains at least
$H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}$
points where the function S(t) changes sign. Earlier similar results were obtained by Atle Selberg for the case
$H\geq T^{{\frac {1}{2}}+\varepsilon }.$
Representationsedit
Dirichlet series^{[29]}edit
An extension of the area of convergence can be obtained by rearranging the original series. The series
in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have
These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.
Theta functionsedit
The Riemann zeta function can be given by a Mellin transform^{[31]}
However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1:
The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development is then^{[32]}
This can be used recursively to extend the Dirichlet series definition to all complex numbers.
The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on x^{s − 1}; that context gives rise to a series expansion in terms of the falling factorial.^{[33]}
This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.)
Globally convergent seriesedit
A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + 2πi/ln 2n for some integer n, was conjectured by Konrad Knopp in 1926 ^{[34]} and proven by Helmut Hasse in 1930^{[35]} (cf. Euler summation):
in the same publication.^{[35]} Research by Iaroslav Blagouchine^{[37]}^{[34]}
has found that a similar, equivalent series was published by Joseph Ser in 1926.^{[38]}
In 1997 K. Maślanka gave another globally convergent (except s = 1) series for the Riemann zeta function:
Here $B_{n}$ are the Bernoulli numbers and $(x)_{k}$ denotes the Pochhammer symbol.^{[39]}^{[40]}
Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points $s=2,4,6,\ldots$, i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.^{[41]}
The asymptotic behavior of the coefficients $A_{k}$ is rather curious: for growing $k$ values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as $k^{-2/3}$). Using the saddle point method, we can show that
On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.^{[43]}^{[44]}^{[45]} Namely, if we define the coefficients $c_{k}$ as
Certain linear combinations of Dirichlet series whose coefficients are terms of the Thue-Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 ^{[50]}). For instance:
There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function
which coincides with the Riemann zeta function when z = 1.
The Clausen functionCl_{s}(θ) can be chosen as the real or imaginary part of Li_{s}(e^{iθ}).
One can analytically continue these functions to the n-dimensional complex space. The special values taken by these functions at positive integer arguments are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.
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Riemann Zeta Function, in Wolfram Mathworld — an explanation with a more mathematical approach
Tables of selected zeros Archived 17 May 2009 at the Wayback Machine
Prime Numbers Get Hitched A general, non-technical description of the significance of the zeta function in relation to prime numbers.
X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary.
Formulas and identities for the Riemann Zeta function functions.wolfram.com
Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23.2 of Abramowitz and Stegun
Frenkel, Edward. "Million Dollar Math Problem" (video). Brady Haran. Archived from the original on 11 December 2021. Retrieved 11 March 2014.
Mellin transform and the functional equation of the Riemann Zeta function—Computational examples of Mellin transform methods involving the Riemann Zeta Function
Visualizing the Riemann zeta function and analytic continuation a video from 3Blue1Brown