# Riemann zeta function

Riemann zeta function_table_infobox_0

∖ { 1 }

{\displaystyle \mathbb {C} \setminus \{1\}}Riemann zeta function_cell_0_2_1

{\displaystyle \mathbb {C} }Riemann zeta function_cell_0_3_1

1 2

{\displaystyle -{\frac {1}{2}}}Riemann zeta function_cell_0_5_1

Limit to +∞Riemann zeta function_header_cell_0_6_0 1

{\displaystyle 1}Riemann zeta function_cell_0_6_1

Value at

2

π

2

6

{\displaystyle {\frac {\pi ^{2}}{6}}}Riemann zeta function_cell_0_7_1

Value at

− 1

1 12

{\displaystyle -{1 \over 12}}Riemann zeta function_cell_0_8_1

Value at

− 2

0

{\displaystyle 0}Riemann zeta function_cell_0_9_1

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 Dirichlet series Riemann zeta function_sentence_0

which converges when the real part of s is greater than 1. Riemann zeta function_sentence_1

More general representations of ζ(s) for all s are given below. Riemann zeta function_sentence_2

The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Riemann zeta function_sentence_3

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. Riemann zeta function_sentence_4

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. Riemann zeta function_sentence_5

The values of the Riemann zeta function at even positive integers were computed by Euler. Riemann zeta function_sentence_6

The first of them, ζ(2), provides a solution to the Basel problem. Riemann zeta function_sentence_7

In 1979 Roger Apéry proved the irrationality of ζ(3). Riemann zeta function_sentence_8

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Riemann zeta function_sentence_9

Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. Riemann zeta function_sentence_10

## Definition Riemann zeta function_section_0

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. Riemann zeta function_sentence_11

(The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) Riemann zeta function_sentence_12

where Riemann zeta function_sentence_13

is the gamma function. Riemann zeta function_sentence_14

In the case σ > 1, the integral for ζ(s) always converges, and can be simplified to the following infinite series: Riemann zeta function_sentence_15

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Riemann zeta function_sentence_16

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. Riemann zeta function_sentence_17

For s = 1, the series is the harmonic series which diverges to +∞, and Riemann zeta function_sentence_18

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. Riemann zeta function_sentence_19

## Specific values Riemann zeta function_section_1

Main article: Particular values of the Riemann zeta function Riemann zeta function_sentence_20

For any positive even integer 2n: Riemann zeta function_sentence_21

where B2n is the 2nth Bernoulli number. Riemann zeta function_sentence_22

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. Riemann zeta function_sentence_23

For nonpositive integers, one has Riemann zeta function_sentence_24

for n ≥ 0 (using the convention that B1 = −1/2). Riemann zeta function_sentence_25

In particular, ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. Riemann zeta function_sentence_26

These are the so-called "trivial zeros" of the zeta function. Riemann zeta function_sentence_27

Via analytic continuation, one can show that: Riemann zeta function_sentence_28

Riemann zeta function_description_list_0

Riemann zeta function_description_list_1

• Riemann zeta function_item_1_2
• Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + ⋯.Riemann zeta function_item_1_3

Riemann zeta function_description_list_2

• Riemann zeta function_item_2_4
• This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.Riemann zeta function_item_2_5

Riemann zeta function_description_list_3

• Riemann zeta function_item_3_6
• 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.Riemann zeta function_item_3_7

Riemann zeta function_description_list_4

• Riemann zeta function_item_4_8
• 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?Riemann zeta function_item_4_9

Riemann zeta function_description_list_5

• Riemann zeta function_item_5_10

Riemann zeta function_description_list_6

## Euler product formula Riemann zeta function_section_2

In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity Riemann zeta function_sentence_29

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): Riemann zeta function_sentence_30

Both sides of the Euler product formula converge for Re(s) > 1. Riemann zeta function_sentence_31

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Riemann zeta function_sentence_32

Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes ∏p p/p − 1) implies that there are infinitely many primes. Riemann zeta function_sentence_33

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Riemann zeta function_sentence_34

Intuitively, the probability that any single number is divisible by a prime (or any integer) p is 1/p. Riemann zeta function_sentence_35

Hence the probability that s numbers are all divisible by this prime is 1/p, and the probability that at least one of them is not is 1 − 1/p. Riemann zeta function_sentence_36

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). Riemann zeta function_sentence_37

Thus the asymptotic probability that s numbers are coprime is given by a product over all primes, Riemann zeta function_sentence_38

(More work is required to derive this result formally.) Riemann zeta function_sentence_39

## Riemann's functional equation Riemann zeta function_section_3

The zeta function satisfies the functional equation: Riemann zeta function_sentence_40

where Γ(s) is the gamma function. Riemann zeta function_sentence_41

This is an equality of meromorphic functions valid on the whole complex plane. Riemann zeta function_sentence_42

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. Riemann zeta function_sentence_43

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). Riemann zeta function_sentence_44

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. Riemann zeta function_sentence_45

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. Riemann zeta function_sentence_46

An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (alternating zeta function): Riemann zeta function_sentence_47

Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e. Riemann zeta function_sentence_48

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). Riemann zeta function_sentence_49

Riemann also found a symmetric version of the functional equation applying to the xi-function: Riemann zeta function_sentence_50

which satisfies: Riemann zeta function_sentence_51

(Riemann's original ξ(t) was slightly different.) Riemann zeta function_sentence_52

## Zeros, the critical line, and the Riemann hypothesis Riemann zeta function_section_4

Main article: Riemann hypothesis Riemann zeta function_sentence_53

The functional equation shows that the Riemann zeta function has zeros at −2, −4,…. Riemann zeta function_sentence_54

These are called the trivial zeros. Riemann zeta function_sentence_55

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. Riemann zeta function_sentence_56

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. Riemann zeta function_sentence_57

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. Riemann zeta function_sentence_58

The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. Riemann zeta function_sentence_59

In the theory of the Riemann zeta function, the set {s ∈ ℂ : Re(s) = 1/2} is called the critical line. Riemann zeta function_sentence_60

For the Riemann zeta function on the critical line, see Z-function. Riemann zeta function_sentence_61

### The Hardy–Littlewood conjectures Riemann zeta function_section_5

In 1914, Godfrey Harold Hardy proved that ζ (1/2 + it) has infinitely many real zeros. Riemann zeta function_sentence_62

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. Riemann zeta function_sentence_63

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]. Riemann zeta function_sentence_64

These two conjectures opened up new directions in the investigation of the Riemann zeta function. Riemann zeta function_sentence_65

### Zero-free region Riemann zeta function_section_6

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. Riemann zeta function_sentence_66

The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line. Riemann zeta function_sentence_67

A better result that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever |t| ≥ 3 and Riemann zeta function_sentence_68

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. Riemann zeta function_sentence_69

### Other results Riemann zeta function_section_7

It is known that there are infinitely many zeros on the critical line. Riemann zeta function_sentence_70

Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then Riemann zeta function_sentence_71

The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. Riemann zeta function_sentence_72

(The Riemann hypothesis would imply that this proportion is 1.) Riemann zeta function_sentence_73

In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + 14.13472514…i (OEIS: ). Riemann zeta function_sentence_74

The fact that Riemann zeta function_sentence_75

for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Riemann zeta function_sentence_76

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. Riemann zeta function_sentence_77

## Various properties Riemann zeta function_section_8

For sums involving the zeta-function at integer and half-integer values, see rational zeta series. Riemann zeta function_sentence_78

### Reciprocal Riemann zeta function_section_9

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n): Riemann zeta function_sentence_79

for every complex number s with real part greater than 1. Riemann zeta function_sentence_80

There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series. Riemann zeta function_sentence_81

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. Riemann zeta function_sentence_82

### Universality Riemann zeta function_section_10

The critical strip of the Riemann zeta function has the remarkable property of universality. Riemann zeta function_sentence_83

This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Riemann zeta function_sentence_84

Since holomorphic functions are very general, this property is quite remarkable. Riemann zeta function_sentence_85

The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975. Riemann zeta function_sentence_86

More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions. Riemann zeta function_sentence_87

### Estimates of the maximum of the modulus of the zeta function Riemann zeta function_section_11

Let the functions F(T;H) and G(s0;Δ) be defined by the equalities Riemann zeta function_sentence_88

Here T is a sufficiently large positive number, 0 < H ≪ ln ln T, s0 = σ0 + iT, 1/2 ≤ σ0 ≤ 1, 0 < Δ < 1/3. Riemann zeta function_sentence_89

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. Riemann zeta function_sentence_90

The case H ≫ ln ln T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial. Riemann zeta function_sentence_91

Anatolii Karatsuba proved, in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates Riemann zeta function_sentence_92

hold, where c1 and c2 are certain absolute constants. Riemann zeta function_sentence_93

### The argument of the Riemann zeta function Riemann zeta function_section_12

The function Riemann zeta function_sentence_94

is called the argument of the Riemann zeta function. Riemann zeta function_sentence_95

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. Riemann zeta function_sentence_96

There are some theorems on properties of the function S(t). Riemann zeta function_sentence_97

Among those results are the mean value theorems for S(t) and its first integral Riemann zeta function_sentence_98

on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for Riemann zeta function_sentence_99

contains at least Riemann zeta function_sentence_100

points where the function S(t) changes sign. Riemann zeta function_sentence_101

Earlier similar results were obtained by Atle Selberg for the case Riemann zeta function_sentence_102

## Representations Riemann zeta function_section_13

### Dirichlet series Riemann zeta function_section_14

An extension of the area of convergence can be obtained by rearranging the original series. Riemann zeta function_sentence_103

The series Riemann zeta function_sentence_104

converges for Re(s) > 0, while Riemann zeta function_sentence_105

converges even for Re(s) > −1. Riemann zeta function_sentence_106

In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k. Riemann zeta function_sentence_107

### Theta functions Riemann zeta function_section_16

The Riemann zeta function can be given by a Mellin transform Riemann zeta function_sentence_108

in terms of Jacobi's theta function Riemann zeta function_sentence_109

However, this integral only converges if the real part of s is greater than 1, but it can be regularized. Riemann zeta function_sentence_110

This gives the following expression for the zeta function, which is well defined for all s except 0 and 1: Riemann zeta function_sentence_111

### Integral Riemann zeta function_section_18

For all s ∈ C, s ≠ 1, the integral relation (cf. Riemann zeta function_sentence_112

Abel–Plana formula) Riemann zeta function_sentence_113

holds true, which may be used for a numerical evaluation of the zeta-function. Riemann zeta function_sentence_114

### Rising factorial Riemann zeta function_section_19

Another series development using the rising factorial valid for the entire complex plane is Riemann zeta function_sentence_115

This can be used recursively to extend the Dirichlet series definition to all complex numbers. Riemann zeta function_sentence_116

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; that context gives rise to a series expansion in terms of the falling factorial. Riemann zeta function_sentence_117

### Hadamard product Riemann zeta function_section_20

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion Riemann zeta function_sentence_118

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler–Mascheroni constant. Riemann zeta function_sentence_119

A simpler infinite product expansion is Riemann zeta function_sentence_120

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. Riemann zeta function_sentence_121

(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.) Riemann zeta function_sentence_122

### Globally convergent series Riemann zeta function_section_21

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 and proven by Helmut Hasse in 1930 (cf. Riemann zeta function_sentence_123

Euler summation): Riemann zeta function_sentence_124

The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994. Riemann zeta function_sentence_125

Hasse also proved the globally converging series Riemann zeta function_sentence_126

in the same publication. Riemann zeta function_sentence_127

Research by Iaroslav Blagouchine has found that a similar, equivalent series was published by Joseph Ser in 1926. Riemann zeta function_sentence_128

Other similar globally convergent series include Riemann zeta function_sentence_129

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. Riemann zeta function_sentence_130

### Series representation at positive integers via the primorial Riemann zeta function_section_22

Here pn# is the primorial sequence and Jk is Jordan's totient function. Riemann zeta function_sentence_131

### Series representation by the incomplete poly-Bernoulli numbers Riemann zeta function_section_23

The function ζ can be represented, for Re(s) > 1, by the infinite series Riemann zeta function_sentence_132

where k ∈ {−1, 0}, Wk is the kth branch of the Lambert W-function, and B n, ≥2 is an incomplete poly-Bernoulli number. Riemann zeta function_sentence_133

## Applications Riemann zeta function_section_27

The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law). Riemann zeta function_sentence_134

Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. Riemann zeta function_sentence_135

In one notable example, the Riemann zeta-function shows up explicitly in one method of calculating the Casimir effect. Riemann zeta function_sentence_136

The zeta function is also useful for the analysis of dynamical systems. Riemann zeta function_sentence_137

### Infinite series Riemann zeta function_section_28

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants. Riemann zeta function_sentence_138

In fact the even and odd terms give the two sums Riemann zeta function_sentence_139

and Riemann zeta function_sentence_140

Parametrized versions of the above sums are given by Riemann zeta function_sentence_141

and Riemann zeta function_sentence_142

where Im denotes the imaginary part of a complex number. Riemann zeta function_sentence_143

There are yet more formulas in the article Harmonic number. Riemann zeta function_sentence_144

## Generalizations Riemann zeta function_section_29

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. Riemann zeta function_sentence_145

These include the Hurwitz zeta function Riemann zeta function_sentence_146

(the convergent series representation was given by Helmut Hasse in 1930, cf. Riemann zeta function_sentence_147

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. Riemann zeta function_sentence_148

For other related functions see the articles zeta function and L-function. Riemann zeta function_sentence_149

The polylogarithm is given by Riemann zeta function_sentence_150

which coincides with the Riemann zeta function when z = 1. Riemann zeta function_sentence_151

The Lerch transcendent is given by Riemann zeta function_sentence_152

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). Riemann zeta function_sentence_153

The Clausen function Cls(θ) that can be chosen as the real or imaginary part of Lis(e). Riemann zeta function_sentence_154

The multiple zeta functions are defined by Riemann zeta function_sentence_155

One can analytically continue these functions to the n-dimensional complex space. Riemann zeta function_sentence_156

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. Riemann zeta function_sentence_157