# Riemann zeta function

The Riemann zeta function represented in a rectangular region of the complex plane. It is generated as a Matplotlib plot using a version of the Domain coloring method.[1]
A view of the Riemann zeta function showing the pole , and two zeros on the critical line.

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series

for when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. 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.[2]

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.

## Definition

Bernhard Riemann's article on the number of primes below a given magnitude.

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 ζ-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:

It can also be defined by the integral

where Γ(s) is the gamma function.

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 Res > 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

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

## Specific values

Riemann zeta function for real s > 1

For any positive even integer 2n:

where B2n is a Bernoulli number.

For negative integers, one has

for n ≥ 1 (using the NIST convention that B1 = −1/2), so in particular ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. 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.

Via analytic continuation, one can show that

gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + · · ·, which can be useful in certain contexts such as string theory.[4]
()
This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.[5]
if we approach from numbers larger than 1. Then this is the harmonic series. But its Cauchy principal value
exists which is the Euler–Mascheroni constant .
()
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]
()
This is called Apéry's constant.
()
This appears when integrating Planck's law to derive the Stefan–Boltzmann law in physics.

## Euler product formula

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

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 ) implies that there are infinitely many primes.[7]

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,

(More work is required to derive this result formally.)[8]

## The functional equation

The Riemann zeta function satisfies the functional equation (known as the Riemann functional equation or Riemann's functional equation)

where Γ(s) is the gamma function, which is an equality of meromorphic functions valid on the whole complex plane. This equation relates values of the Riemann zeta function at the points s and 1 s. 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 these are known as the trivial zeros of ζ(s). When s is an even positive integer, the product
sin(πs/2)Γ(1s) on the right is regular and non-zero because Γ(1s) has a simple pole: the functional equation thus relates the values of the Riemann zeta function at odd negative integers and even positive integers.

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)

Incidentally, this relation is interesting also because it actually exhibits ζ(s) as a Dirichlet series (of the η-function) which is convergent (albeit non-absolutely) in the larger half-plane σ > 0 (not just σ > 1), up to an elementary factor.

Riemann also found a symmetric version of the functional equation (which he assigned the letter ξ [small xi]), given by first defining

The functional equation is then given by

(Riemann defined a similar but different function which he called ξ(t).)

## Zeros, the critical line, and the Riemann hypothesis

Main article: Riemann hypothesis
Apart from the trivial zeros, the Riemann zeta function doesn't have any zero on the right of σ=1 and on 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 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 has infinitely many real zeros.

Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of on intervals of large positive real numbers. In the following, is the total number of real zeros and the total number of zeros of odd order of the function lying in the interval .

1. For any , there exists a such that when and , the interval contains a zero of odd order.
2. For any , there exists a and such that the inequality holds when and .

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 (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 percentage of the nontrivial zeros lies on the critical line.

In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... (). The fact that 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.

### Reciprocal

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

for every complex number s with real part > 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 very first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975,[11] which results non-applicable to solve Riemann's Hypothesis. A latter and more accurate proof was given in 2003 by R. Garunkštis. In 2015, A. Quartarone gave an even more accurate and general proof of this theorem. Nevertheless, none of the proofs above are definitive or complete, since a definitive proof would imply that Riemann's Hypothesis is true.

### Estimates of the maximum of the modulus of the zeta function

Let the functions and be defined by the equalities

Here is a sufficiently large positive number, , , , . Estimating the values and from below shows, how large (in modulus) values can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip .

The case was studied by Ramachandra; the case , where is a sufficiently large constant, is trivial.

Karatsuba proved,[12][13] in particular, that if the values and exceed certain sufficiently small constants, then the estimates

hold, where are certain absolute constants.

### The argument of the Riemann zeta-function

The function is called the argument of the Riemann zeta function. Here is the increment of an arbitrary continuous branch of along the broken line joining the points and There are some theorems on properties of the function . Among those results[14][15] are the mean value theorems for and its first integral on intervals of the real line, and also the theorem claiming that every interval for contains at least

points where the function changes sign. Earlier similar results were obtained by Atle Selberg for the case .

## Representations

### Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series (Konrad Knopp, Theory of Functions, 1945, p. 51-55). The series

converges for , while

converges even for . In this way, the area of convergence can be extended to for any .

### Mellin transform

The Mellin transform of a function ƒ(x) is defined as

in the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we have

where Γ denotes the Gamma function. By modifying the contour, Riemann showed that

for all s, where the contour C starts and ends at +∞ and circles the origin once.

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, then

for values with Re(s) > 1.

A similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers pn with a weight of 1/n, so that

Now 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 functions

The Riemann zeta function can be given formally by a divergent Mellin transform[16]

in terms of Jacobi's theta function

However this integral does not converge for any value of s and so needs to be regularized: this gives the following expression for the zeta function:

### Laurent series

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 then is

The constants γn here are called the Stieltjes constants and can be defined by the limit

The constant term γ0 is the Euler–Mascheroni constant.

### Integral

For all the integral relation (cf. Abel–Plana formula)

holds true, which may be used for a numerical evaluation of the zeta-function.[17]

### Rising factorial

Another series development using the rising factorial valid for the entire complex plane is

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 xs1; that context gives rise to a series expansion in terms of the falling factorial.[18]

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

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 zeroes, i.e. the factors for a pair of zeroes of the form ρ and 1  ρ should be combined.)

### Logarithmic derivative on the critical strip

where is the density of zeros of ζ on the critical strip 0 < Re(s) < 1 (δ is the Dirac delta distribution, and the sum is over the nontrivial zeros ρ of ζ).

### Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers s except for some integer n, was conjectured by Konrad Knopp and proven by Helmut Hasse in 1930 (cf. Euler summation):

The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).

Hasse also proved the globally converging series

in the same publication. However, recent research indicates that the latter series was discovered earlier, by Joseph Ser in 1926.[19]

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.

### Series representation at positive integers via the primorial

Here pn# is the primorial sequence and Jk is Jordan's totient function.[20]

### Series representation by the incomplete poly-Bernoulli numbers

The function can be represented, for , by the infinite series

where , is the -th branch of the Lambert W function, and is an incomplete poly-Bernoulli number.[21]

## Applications

The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law).

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 the calculation of the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.[22]

### Infinite series

The zeta function evaluated at positive integers appears in infinite series representations of a number of constants.[23] There are more formulas in the article Harmonic number.

In fact the even and odd terms give the two sums     and
where γ is Euler's constant.
where   represents the imaginary part of a complex number.

Some zeta series evaluate to more complicated expressions

for   .

## Generalizations

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,[24] 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.

The polylogarithm is given by

which coincides with the Riemann zeta function when z = 1.

The Lerch transcendent is given by

which coincides with the Riemann zeta function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).

The Clausen function Cls(θ) that can be chosen as the real or imaginary part of Lis(e ).

The multiple zeta functions are defined by

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.

## Notes

1. http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb
2. 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.Bombieri, Enrico. "The Riemann Hypothesis – official problem description" (PDF). Clay Mathematics Institute. Retrieved 2014-08-08.
3. Devlin, Keith (2002). The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. New York: Barnes & Noble. pp. 43–47. ISBN 978-0-7607-8659-8.
4. Polchinski, Joseph (1998). String Theory, Volume I: An Introduction to the Bosonic String. Cambridge University Press. p. 22. ISBN 978-0-521-63303-1.
5. A J Kainz and U M Titulaer, An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations, pp. 1855-1874, J. Phys. A: Mathem. and General, V 25, No 7, 1992
6. C. S. Ogilvy & J. T. Anderson Excursions in Number Theory, pp. 29–35, Dover Publications Inc., 1988 ISBN 0-486-25778-9
7. Charles Edward Sandifer, How Euler did it, The Mathematical Association of America, 2007, p. 193. ISBN 978-0-88385-563-8
8. J. E. Nymann (1972). "On the probability that k positive integers are relatively prime". Journal of Number Theory. 4 (5): 469–473. Bibcode:1972JNT.....4..469N. doi:10.1016/0022-314X(72)90038-8.
9. Diamond, Harold G. (1982), "Elementary methods in the study of the distribution of prime numbers", Bulletin of the American Mathematical Society, 7 (3): 553–89, doi:10.1090/S0273-0979-1982-15057-1, MR 670132.
10. 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.
11. Voronin, S.M. (1975) "Theorem on the Universality of the Riemann Zeta Function." Izv. Akad. Nauk SSSR, Ser. Matem. 39 pp.475-486. Reprinted in Math. USSR Izv. 9, 443-445, 1975
12. 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.
13. 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.
14. Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60:3): 448–449.
15. Karatsuba, A. A. (1996). "On the function S(t)". Izv. Ross. Akad. Nauk, Ser. Mat. (60:5): 27–56.
16. Neukirch, Jürgen (1999). Algebraic number theory. Springer. p. 422. ISBN 3-540-65399-6.
17. Mathematik-Online-Kurs: Numerik – Numerische Integration der Riemannschen Zeta-Funktion
18. http://linas.org/math/poch-zeta.pdf
19. Blagouchine, Iaroslav V. (2016), "Expansions of generalized Euler's constants into the series of polynomials in π2 and into the formal enveloping series with rational coefficients only", Journal of Number Theory, 158: 365–396, arXiv:, doi:10.1016/j.jnt.2015.06.012.
20. Mező, István (2013). "The primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
21. Komatsu, Takao; Mező, István (2016). "Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers". Publicationes Mathematicae Debrecen. 88 (3-4): 357-368.
22. Unless otherwise noted, the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
23. Hasse, Helmut (1930). "Ein Summierungsverfahren für die Riemannsche ζ-Reihe". Mathematische Zeitschrift. 32 (1): 458–464. doi:10.1007/BF01194645