Riemann zeta function
Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in pure mathematics.
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.
The Riemann zeta function ζ(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.) When Re(s) = σ > 1, the function can be written as a converging summation or integral:
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
Euler's product formula
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 Πp p/p − 1) implies that there are infinitely many primes.
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/ps, and the probability that at least one of them is not is 1 − 1/ps. 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,
Riemann's functional equation
This zeta function satisfies the functional equation
A proof of the functional equation proceeds as follows: We observe that if , then
As a result, if then
For convenience, let
By the Poisson summation formula we have
This is equivalent to
which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1 − s. Hence
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):
Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e.
Riemann also found a symmetric version of the functional equation applying to the xi-function:
(Riemann's original ξ(t) was slightly different.)
The 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 hypothesis
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 , which is called the critical strip. The set 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.
For the Riemann zeta function on the critical line, see Z-function.
|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|
Number of zeros in the critical strip
Let be the number of zeros of in the critical strip , whose imaginary parts are in the interval . Trudgian proved that, if , then
The Hardy–Littlewood conjectures
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 N0(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 T0(ε) > 0 such that when
- For any ε > 0, there exists a T0(ε) > 0 and cε > 0 such that the inequality
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
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. A better result that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever and |t| ≥ 3.
In 2015, Mossinghoff and Trudgian proved that zeta has no zeros in the region
for |t| ≥ 2. This is the largest known zero-free region in the critical strip for .
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.
It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) 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.)
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 a line with real part 1.
For any positive even integer 2n,
For nonpositive integers, one has
Via analytic continuation, one can show that
The demonstration of the particular value
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.
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. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions.
Estimates of the maximum of the modulus of the zeta function
Let the functions F(T;H) and G(s0;Δ) be defined by the equalities
Here T is a sufficiently large positive number, 0 < H ≪ log log T, s0 = σ0 + iT, 1/2 ≤ σ0 ≤ 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.
hold, where c1 and c2 are certain absolute constants.
The argument of the Riemann zeta function
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.
on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for
contains at least
points where the function S(t) changes sign. Earlier similar results were obtained by Atle Selberg for the case
An extension of the area of convergence can be obtained by rearranging the original series. The series
converges for Re(s) > 0, while
converges even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k.
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
- and ,
for all s (where H denotes the Hankel contour).
for values with Re(s) > 1.
A similar Mellin transform involves the Riemann function J(x), which counts prime powers pn with a weight of 1/n, so that
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.
The Riemann zeta function can be given by a Mellin transform
in terms of Jacobi's theta function
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 constant term γ0 is the Euler–Mascheroni constant.
For all s ∈ C, s ≠ 1, the integral relation (cf. Abel–Plana formula)
holds true, which may be used for a numerical evaluation of the zeta function.
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 xs − 1; that context gives rise to a series expansion in terms of the falling factorial.
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 in 1926  and proven by Helmut Hasse in 1930 (cf. Euler summation):
The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.
Hasse also proved the globally converging series
Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.
Series representation at positive integers via the primorial
Series representation by the incomplete poly-Bernoulli numbers
The function ζ can be represented, for Re(s) > 1, by the infinite series
The Mellin transform of the Engel map
The Mellin transform of the map is related to the Riemann zeta function by the formula
A classical algorithm, in use prior to about 1930, proceeds by applying the Euler-Maclaurin formula to obtain, for n and m positive integers,
where, letting denote the indicated Bernoulli number,
and the error satisfies
with σ = Re(s).
A modern numerical algorithm is the Odlyzko–Schönhage algorithm.
Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann zeta function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.
In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of , the value of
The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.
In fact the even and odd terms give the two sums
Parametrized versions of the above sums are given by
all of which are continuous at . Other sums include
where Im denotes the imaginary part of a complex number.
There are yet more formulas in the article Harmonic number.
(the convergent series representation was given by Helmut Hasse in 1930, cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (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.
The polylogarithm is given by
which coincides with the Riemann zeta function when z = 1. The Clausen function Cls(θ) can be chosen as the real or imaginary part of Lis(eiθ).
The Lerch transcendent is given by
which coincides with the Riemann zeta function when z = 1 and q = 1 (the lower limit of summation in the Lerch transcendent is 0, not 1).
The multiple zeta functions are defined by
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.
- 1 + 2 + 3 + 4 + ···
- Arithmetic zeta function
- Generalized Riemann hypothesis
- Lehmer pair
- Prime zeta function
- Riemann Xi function
- Riemann–Siegel theta function
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