Riemann Zeta Function

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis.

In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859).

is implemented in the Wolfram Language as Zeta[s].

These additional surfaces and contours show the real part, imaginary part, and complex modulus of over a wider portion of the complex plane.

The zero contours of (red) and (blue) meet at the Riemann zeta function zeros, marked by dots. The assertion that every nontrivial zero lies on the critical line is the Riemann hypothesis.

The plot above shows the "ridges" of for and . Saidak and Zvengrowski (2003, p. 151) conjectured that, for each fixed , is strictly decreasing in for . This conjecture implies the Riemann hypothesis (Borwein and Bailey 2003, pp. 95-96).

On the real line with , the Riemann zeta function can be defined by the integral

where is the gamma function. If is an integer , then the following identity holds

so

To evaluate , let so that and plug in the above identity to obtain

Integrating the final expression in (8) gives , which cancels the factor and gives the most common form of the Riemann zeta function,

which is sometimes known as a p-series.

The Riemann zeta function can also be defined in terms of multiple integrals by

and as a Mellin transform by

for , where is the fractional part (Balazard and Saias 2000).

It appears in the unit square integral

valid for (Guillera and Sondow 2005). For a nonnegative integer, this formula is due to Hadjicostas (2002), and the special cases and are due to Beukers (1979).

Note that the zeta function has a singularity at , where it reduces to the divergent harmonic series.

The Riemann zeta function satisfies the reflection functional equation

(Hardy 1999, p. 14; Krantz 1999, p. 160), a similar form of which was conjectured by Euler for real (Euler, read in 1749, published in 1768; Ayoub 1974; Havil 2003, p. 193). A symmetrical form of this functional equation is given by

(Ayoub 1974), which was proved by Riemann for all complex (Riemann 1859). Both forms are interpreted as meromorphic identities, with limiting values at apparent singularities.

As defined above, the zeta function with a complex number is defined for . However, has a unique analytic continuation to the entire complex plane, excluding the point , which corresponds to a simple pole with complex residue 1 (Krantz 1999, p. 160). In particular, as , obeys

where is the Euler-Mascheroni constant (Whittaker and Watson 1990, p. 271).

To perform the analytic continuation for , first write for

so rewriting in terms of immediately gives

Therefore,

Here, the sum on the right-hand side is exactly the Dirichlet eta function (sometimes also called the alternating zeta function). The series converges in the right half-plane . At with nonzero integer , the quotient is interpreted by its removable limit. Then equation (13) can be used to analytically continue it to the rest of the complex plane. Analytic continuation can also be performed using a contour integral over a Hankel contour. A globally convergent series for the Riemann zeta function (which provides the analytic continuation of to the entire complex plane except ) is given by

(Havil 2003, p. 206), where is a binomial coefficient. The same removable-limit convention applies when and . The formula was conjectured by Knopp around 1930, proved by Hasse (1930), and rediscovered by Sondow (1994). This equation is related to renormalization and random variates (Biane et al. 2001) and can be derived by applying Euler's series transformation to the Dirichlet eta function series.

Hasse (1930) also proved the related globally (but more slowly) convergent series

that, unlike (21), can also be extended to a generalization of the Riemann zeta function known as the Hurwitz zeta function . is defined such that

(If the singular term is excluded from the sum definition of , then as well.) Expanding about gives

where are the so-called Stieltjes constants.

The Riemann zeta function can also be defined in the complex plane by the contour integral

for , with limiting values at integers where has a pole. The contour is illustrated above (Havil 2003, pp. 193 and 249-252).

Zeros of come in (at least) two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" at certain in the "critical strip" . The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of all have real part , a line called the "critical line." This is now known to be true for the first roots.

A weaker assertion is the quasi-Riemann hypothesis, which asks that the Riemann zeta function zeros lie outside a fixed half-plane for some real number . OpenAI (2026) announced such a region with , with the pole at excluded. This claim does not establish the Riemann hypothesis.

The plot above shows the real part and imaginary part of (i.e., values of along the critical line) as is varied from 0 to 35 (Derbyshire 2004, p. 221).

The Riemann zeta function can be split up into

where and are the Riemann-Siegel functions.

The Riemann zeta function is related to the Dirichlet lambda function and Dirichlet eta function by

and

(Spanier and Oldham 1987). The quotients are interpreted by limits where their denominators vanish, apart from the pole at .

For , it is related to the Liouville function by

(Lehman 1960, Hardy and Wright 1979). In the same domain,

where is the number of distinct prime factors of (Hardy and Wright 1979, p. 254).

For a negative even integer , , ...,

giving the first few as

with rational coefficients whose numerators and denominators are given by OEIS A117972 and OEIS A117973, respectively. At ,

where is the Glaisher-Kinkelin constant. Using equation (21) gives the derivative

which can be derived directly from the Wallis formula (Sondow 1994). can also be derived directly from the Euler-Maclaurin summation formula (Edwards 2001, pp. 134-135). For positive integers , can be expressed using , powers of , the Euler-Mascheroni constant , the Stieltjes constants , and values with . The first few examples are

Derivatives can also be given in closed form, for example,

(OEIS A114875).

The derivative of the Riemann zeta function for is defined by

The series for about is

where are Stieltjes constants.

The following table lists values of for small positive integers .

In 1739, Euler found the rational coefficients in in terms of the Bernoulli numbers. Combined with the 1882 proof by Lindemann that is transcendental, this proves that is transcendental. The study of is significantly more difficult. Apéry (1979) proved to be irrational. As a result of Apéry's important discovery, is sometimes called Apéry's constant. Rivoal (2000) and Ball and Rivoal (2001) proved that there are infinitely many integers such that is irrational, and subsequently that at least one of , , ..., is irrational (Rivoal 2001). This result was subsequently tightened by Zudilin (2001), who showed that at least one of , , , or is irrational. Fauzan (2026) announced a proof that zeta(5) is irrational.

Lai (2026) reported that 1, Apéry's constant, and are linearly independent over the rationals.

A number of interesting sums for , with a positive integer, can be written in terms of binomial coefficients as the binomial sums

(Guy 1994, p. 257; Bailey et al. 2007, p. 70). Apéry arrived at his result with the aid of the sum formula above. A relation of the form

has been searched for with a rational or algebraic number, but if is a root of a polynomial of degree 25 or less, then the Euclidean norm of the coefficients must be larger than , and if is algebraic of degree 25 or less, then the norm of coefficients must exceed (Bailey et al. 2007, pp. 70-71, updating Bailey and Plouffe). No analogous formula of this single-series, constant-coefficient form is known for . Multiterm binomial sums for higher odd arguments are given below.

The identity

holds for , with the cotangent quotient interpreted by its limit at , and gives an Apéry-like formula for positive even arguments (Bailey et al. 2007, pp. 72-77). The forms other than the power series extend meromorphically to complex numbers away from nonzero integers. Here is a generalized hypergeometric function.

The Riemann zeta function may be computed analytically for positive integers using either contour integration or Parseval's theorem with the appropriate Fourier series. An unexpected and important formula involving a product over the primes was first discovered by Euler in 1737,

For , multiplication by removes from the Dirichlet series the terms whose indices are divisible by the th prime . After all such factors have been included, only the term with index 1 remains. Therefore,

which is known as the Euler product formula (Hardy 1999, p. 18; Krantz 1999, p. 159), and called "the golden key" by Derbyshire (2004, pp. 104-106). The formula can also be written

where and are the primes congruent to 1 and 3 modulo 4, respectively.

For even ,

where is a Bernoulli number (Mathews and Walker 1970, pp. 50-53; Havil 2003, p. 194). Another intimate connection with the Bernoulli numbers is provided by

for , which can be written

for . (In both cases, only the even cases are of interest since trivially for odd .) Rewriting (62),

for , 3, ... (Havil 2003, p. 194), where is a Bernoulli number. The first few negative-odd zeta values are , 1/120, , 1/240, .... Their numerators are the negatives of OEIS A001067, and their denominators are OEIS A006953.

A harmonic number representation is

where is a harmonic number (Stark 1974). In addition, can be expressed as the sum limit

for , 5, ..., with tending to infinity through positive integers (Apostol 1973, given incorrectly in Stark 1974).

For and the Möbius function,

(Havil 2003, p. 209).

Euler gave to for even (Wells 1986, p. 54), and Stieltjes (1993) determined the values of , ..., to 30 digits of accuracy in 1887. The denominators of the rational numbers for , 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (OEIS A002432). The numbers of decimal digits in the denominators of for , 2, ... are 5, 133, 2277, 32660, 426486, 5264705, ... (OEIS A114474).

For a positive integer , an integral for positive even arguments is given by

and integrals for positive odd integers are given by

where is an Euler polynomial and is a Bernoulli polynomial (Cvijović and Klinowski 2002; J. Crepps, pers. comm., Apr. 2002).

The value of can be computed by performing the inner sum in equation (21) with ,

to obtain

where is the Kronecker delta.

Similarly, the value of can be computed by performing the inner sum in equation (21) with ,

which gives

This value is related to a deep result in renormalization theory (Elizalde et al. 1994, 1995, Bloch 1996, Lepowski 1999).

It is apparently not known if the value

(OEIS A059750) can be expressed in terms of known mathematical constants. This constant appears, for example, in Knuth's series.

Rapidly converging series for for odd were first discovered by Ramanujan (Zucker 1979, 1984, Berndt 1988, Bailey et al. 1997, Cohen 2000). For and ,

where is again a Bernoulli number and is a binomial coefficient. The values of the finite Bernoulli-number sums (divided by ) in (79) for , 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/390769879500, 7708537/21438612514068750, ... (OEIS A057866 and A057867). For and , the corresponding formula is slightly messier,

(Cohen 2000).

Defining

the first few values can then be written

(Plouffe 1998).

The sum (OEIS A255702) also occurs through the combination in series expressions for the torsional rigidity multipliers of a square and an isosceles right triangle.

Another set of related formulas are

(Plouffe 2006).

For odd integers , define the parity-dependent series

The related identities

communicated by L. Kleinwaks (pers. comm., Mar. 19, 2007), follow by separating the even and odd terms in Ramanujan's transformation formula for Lambert series (Berndt and Straub 2017).

Multiterm sums for with odd include

(Borwein and Bradley 1996, 1997; Bailey et al. 2007, p. 71), where is a generalized harmonic number.

A number of sum identities involving include

Sums involving integer multiples of the argument include

where extends the harmonic number to complex arguments, is the digamma function, and the fractional powers of use their principal value.

Two surprising sums involving are given by

where is the Euler-Mascheroni constant (Havil 2003, pp. 109 and 111-112). The first of these identities can be generalized to

(T. Drane, pers. comm., Jul. 7, 2006) for .

Other unexpected sums are

(Tyler and Chernhoff 1985; Boros and Moll 2004, p. 248) and

The first of these identities is a special case of

valid for real and by continuity at , where is a Hurwitz zeta function (Danese 1967; Boros and Moll 2004, p. 248).

Considering the sum

then

where is the natural logarithm of 2, which is a particular case of the identity, valid for ,

where is the digamma function, is the Euler-Mascheroni constant, and the right-hand side is interpreted by its limit at . This identity follows from the series, valid for ,

(B. Cloitre, pers. comm., Dec. 11, 2005; cf. Borwein et al. 2000, eqn. 27).

A generalization of a result of Ramanujan (who gave the case) is given by

where is a nonnegative integer and is a binomial coefficient (B. Cloitre, pers. comm., Sep. 20, 2005).

An additional set of sums over is given by

(OEIS A093720, A076813, and A093721), where is a modified Bessel function of the first kind, is a regularized hypergeometric function. These sums have no known closed-form expression.

The reciprocal , plotted above, is the asymptotic density of pth-powerfree numbers for integers (i.e., squarefree numbers, cubefree numbers, etc.). The following table gives the number of pth-powerfree numbers for several values of .

The van de Lune constant is the sharp positivity threshold for the real part of the Riemann zeta function. For every real , when (van de Lune 1983, Arias de Reyna et al. 2012).

Pańkowski (2026) proves joint universality of the shifts for arbitrary nonzero real parameters with pairwise distinct absolute values. For each , let be a compact set in with connected complement, and let be continuous and nonvanishing on and holomorphic in its interior. For every , the simultaneous inequalities

hold for a set of whose measure divided by has positive lower limit as . The restrictions on the shift parameters are sharp for independent approximation targets.