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 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.
Riemann zeta function
The Riemann zeta function ζ(z) plotted with domain coloring.^{[1]}
The pole at $z=1$, and two zeros on the critical line.
Definition
Bernhard Riemann's article On the number of primes below a given magnitude.
The Riemann zeta functionζ(s) is a function of a complex variable s = σ + it. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.)
The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:
The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.
Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1.^{[3]}
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
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
$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$
for n ≥ 0 (using the NIST 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.
This gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts such as string theory.^{[4]}
$\zeta (0)=-{\tfrac {1}{2}};$
Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + ⋯.
This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.
The demonstration of this equality 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?^{[6]}
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):
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 functional equation
A proof of the functional equation proceeds as follows: We observe that, if $\sigma >0$ $\int \limits _{0}^{\infty }x^{{1 \over 2}{s}-1}e^{-{n}^{2}\pi x}\,dx={\Gamma ({s \over 2}) \over {n^{s}\pi ^{s \over 2}}}$
As a result, if $\sigma >1$ ${\Gamma ({s \over 2})\zeta (s) \over {\pi ^{s \over 2}}}={\sum _{n=1}^{\infty }{\int \limits _{0}^{\infty }x^{{{s} \over 2}-1}e^{-n^{2}\pi x}\,dx}}={\int \limits _{0}^{\infty }x^{{{s} \over 2}-1}{\sum _{n=1}^{\infty }e^{-n^{2}\pi x}\,dx}}$
With the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on $\sigma$)
For convenience, let $\psi (x):=\sum _{n=1}^{\infty }e^{-n^{2}\pi x}$
Then $\zeta (s)={\pi ^{s \over 2} \over \Gamma ({s \over 2})}\int \limits _{0}^{\infty }x^{{1 \over 2}{s}-1}\psi (x)\,dx$
Given that $\sum _{n=-\infty }^{\infty }{e^{-n^{2}\pi x}}={1 \over {\sqrt {x}}}\sum _{n=-\infty }^{\infty }{e^{-n^{2}\pi \over x}}$
Then $2\psi (x)+1={1 \over {\sqrt {x}}}\{2\psi \left({1 \over x}\right)+1\}$
Hence $\pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta (s)=\int \limits _{0}^{1}x^{{{s} \over 2}-1}\psi (x)\,dx+\int \limits _{1}^{\infty }x^{{{s} \over 2}-1}\psi (x)\,dx$
This is equivalent to $\int \limits _{0}^{1}x^{{{s} \over 2}-1}\{{1 \over {\sqrt {x}}}\psi \left({1 \over x}\right)+{1 \over 2{\sqrt {x}}}-{1 \over 2}\}\,dx+\int \limits _{1}^{\infty }x^{{{s} \over 2}-1}\psi (x)\,dx$
Or ${1 \over {s-1}}-{1 \over s}+\int \limits _{0}^{1}x^{{{s} \over 2}-{3 \over 2}}\psi \left({1 \over x}\right)\,dx+\int \limits _{1}^{\infty }x^{{{s} \over 2}-1}\psi (x)\,dx$ $={1 \over {s({s-1})}}+\int \limits _{1}^{\infty }\left({x^{-{{s} \over 2}-{1 \over 2}}+x^{{{s} \over 2}-1}}\right)\psi (x)\,dx$
Which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1-s. Hence $\pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta (s)=\pi ^{-{1 \over 2}+{s \over 2}}\Gamma \left({1 \over 2}-{s \over 2}\right)\zeta ({1-s})$
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 (alternating zeta function):
Zeros, the critical line, and the Riemann hypothesis
Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of σ = 1 and to the left of σ = 0 (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1/2 and, according to the Riemann hypothesis, they all lie on the line σ = 1/2.
This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.
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 impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s ∈ ℂ : 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set {s ∈ ℂ : Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.
The Hardy–Littlewood conjectures
In 1914, Godfrey Harold Hardy proved that ζ(1/2 + it) has infinitely many real zeros.
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 region
The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.^{[9]} A better result^{[10]} that follows from an effective form of Vinogradov's mean-value theorem is that ζ(σ + it) ≠ 0 whenever |t| ≥ 3 and
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 results
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 (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.
Various properties
For sums involving the zeta-function at integer and half-integer values, see rational zeta series.
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.
Universality
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.^{[11]} More recent work has included effective versions of Voronin's theorem^{[12]} and extending it to Dirichlet L-functions.^{[13]}^{[14]}
Estimates of the maximum of the modulus of the zeta function
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 ≪ ln ln 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 ≫ ln ln T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.
Anatolii Karatsuba proved,^{[15]}^{[16]} 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^{[17]}^{[18]} 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
$H\geq T^{{\frac {1}{2}}+\varepsilon }.$.
Representations
Dirichlet series
An extension of the area of convergence can be obtained by rearranging the original series.^{[19]} 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
Starting with the integral formula $\zeta (n){\Gamma (n)}=\int _{0}^{\infty }{\frac {x^{n-1}}{e^{x}-1}}\mathrm {d} x,$ one can show^{[20]} by substitution and iterated differentation for natural $k\geq 2$
using the notation of umbral calculus where each power $\zeta ^{r}$ is to be replaced by $\zeta (r)$, so e.g. for $k=2$ we have $\int _{0}^{\infty }{\frac {x^{n}e^{x}}{(e^{x}-1)^{2}}}\mathrm {d} x={n!}\zeta (n),$ while for $k=4$ this becomes
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 functions
The Riemann zeta function can be given formally by a divergent Mellin transform^{[21]}
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
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.^{[23]}
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 series
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^{[24]} and proven by Helmut Hasse in 1930^{[25]} (cf. Euler summation):
in the same publication,^{[25]} but research by Iaroslav Blagouchine^{[27]} has found that this latter series was actually first published by Joseph Ser in 1926.^{[28]} New proofs for both of these results were offered by Demetrios Kanoussis in 2017.^{[29]}
where k ∈ {−1, 0}, W_{k} is the kth branch of the Lambert W-function, and B^{(μ)} _{n, ≥2} is an incomplete poly-Bernoulli number.^{[32]}
Numerical algorithms
For $v=1,2,3,\dots$ , the Riemann zeta function has for fixed $\sigma _{0}<v$ and for all $\sigma \leq \sigma _{0}$ the following representation in terms of three absolutely and uniformly converging series,^{[33]}
where for positive integer $s=k$ one has to take the limit value $\lim _{s\to k}E_{\mu }\left(s\right)$. The derivatives of $\zeta (s)$ can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision, $\zeta (s)$ and its derivatives using at most $C\left(\epsilon \right)\left|\tau \right|^{{\frac {1}{2}}+\epsilon }$ summands for any $\epsilon >0$, with explicit error bounds. For $\zeta (s)$, these are as follows:
For a given argument $s$ with $0\leq \sigma \leq 2$ and $0<t$ one can approximate $\zeta (s)$ to any accuracy $\delta \leq 0.05$ by summing the first series to $n=\left\lceil 3.151\cdot vN\right\rceil$, $E_{1}\left(s\right)$ to $m=\left\lceil N\right\rceil$ and neglecting $E_{-1}\left(s\right)$, if one chooses $v$ as the next higher integer of the unique solution of $x-\max \left({\frac {1-\sigma }{2}},0\right)\ln \left({\frac {1}{2}}+x+\tau \right)=\ln {\frac {8}{\delta }}$ in the unknown $x$, and from this $N=1.11\left(1+{\frac {{\frac {1}{2}}+\tau }{v}}\right)^{\frac {1}{2}}$. For $t=0$ one can neglect $E_{1}\left(s\right)$ altogether. Under the mild condition $\tau >{\frac {5}{3}}\left({\frac {3}{2}}+\ln {\frac {8}{\delta }}\right)$ one needs at most $2+8{\sqrt {1+\ln {\frac {8}{\delta }}+\max \left({\frac {1-\sigma }{2}},0\right)\ln \left(2\tau \right)}}~{\sqrt {\tau }}$ summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.^{[33]}
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
(the convergent series representation was given by Helmut Hasse in 1930,^{[25]} cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles zeta function and L-function.
One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.
Fractional derivative
In the case of the Riemann zeta function, a difficulty is represented by the fractional differentiation in the complex plane. The Ortigueira generalization of the classical Caputo fractional derivative solves this problem. The α-order fractional derivative of the Riemann zeta function is given by ^{[36]}
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^Ford, K. (2002). "Vinogradov's integral and bounds for the Riemann zeta function". Proc. London Math. Soc. 85 (3): 565–633. doi:10.1112/S0024611502013655.
^Voronin, S. M. (1975). "Theorem on the Universality of the Riemann Zeta Function". Izv. Akad. Nauk SSSR, Ser. Matem. 39: 475–486. Reprinted in Math. USSR Izv. (1975) 9: 443–445.
^Ramūnas Garunkštis; Antanas Laurinčikas; Kohji Matsumoto; Jörn Steuding; Rasa Steuding (2010). "Effective uniform approximation by the Riemann zeta-function". Publicacions Matemàtiques. 54: 209–219. doi:10.1090/S0025-5718-1975-0384673-1. JSTOR43736941.
^Bhaskar Bagchi (1982). "A Joint Universality Theorem for Dirichlet L-Functions". Mathematische Zeitschrift. 181: 319–334. doi:10.1007/bf01161980. ISSN0025-5874.
^Steuding, Jörn (2007). Value-Distribution of L-Functions. Lecture Notes in Mathematics. Berlin: Springer. p. 19. doi:10.1007/978-3-540-44822-8. ISBN 3-540-26526-0.
^Karatsuba, A. A. (2001). "Lower bounds for the maximum modulus of ζ(s) in small domains of the critical strip". Mat. Zametki. 70 (5): 796–798.
^Karatsuba, A. A. (2004). "Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. 68 (8): 99–104.
^Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60): 448–449.
^Karatsuba, A. A. (1996). "On the function S(t)". Izv. Ross. Akad. Nauk, Ser. Mat. 60 (5): 27–56.
^Knopp, Konrad (1945). Theory of Functions. pp. 51–55.
^Blagouchine, Iaroslav V. (2016). "Expansions of generalized Euler's constants into the series of polynomials in π^{−2} and into the formal enveloping series with rational coefficients only". Journal of Number Theory. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012.
^Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
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