P

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 defined over the complex plane for one complex variable, and is conventionally denoted s (instead of the usual z) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). It is implemented in Mathematica as Zeta[s].

The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically is not a coincidence, since monotonic decrease in fact implies the Riemann hypothesis (Zvengrowski and Saidak 2003; Borwein and Borwein 2003, pp. 95-96).

On the real line with x > 1, the Riemann zeta function can be defined by the integral

where is the gamma function. If x is an integer n, then we have the identity

so

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

Integrating the final expression in (6) gives , which cancels the factor and gives the most common form of the Riemann zeta function,

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).

Note that the zeta function has a singularity at n = 1, 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 s (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 s (Riemann 1859).

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 s = 1, 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 , write

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). While this formula defines for only the right half-plane , equation (10) can be used to analytically continue it to the rest of the complex plane. Analytic continuation can also be performed using Hankel functions. A globally convergent series for the Riemann zeta function (which provides the analytic continuation of to the entire complex plane except s = 1) is given by

(Havil 2003, p. 206), where is a binomial coefficient, which 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 with n = 0 to equation (17).

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

that, unlike (18), 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 s = 1 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 all , where 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 s = -2, -4, -6, ..., and "nontrivial zeros" at certain

for s 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.

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

The Riemann zeta function can be split up into

where Z(t) 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). It is related to the Liouville function by

(Lehman 1960, Hardy and Wright 1979). Furthermore,

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

The derivative of the Riemann zeta function for is defined by

Using equation (18) gives the special case

which can be derived directly from the Wallis formula (Sondow 1994). In general, can be expressed analytically in terms of , , the Euler-Mascheroni constant , and the Stieltjes constants , with the first few examples being

The series for about s = 1 is

where are Stieltjes constants.

In 1739, Euler found the rational coefficients C in in terms of the Bernoulli numbers. Which, when combined with the 1882 proof by Lindemann that is transcendental, effectively proves that is transcendental. The study of is significantly more difficult. Apéry (1979) finally proved to be irrational, but no similar results are known for other odd n. 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 n 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 one of , , , or is irrational.

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

(Guy 1994, p. 257). 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 (Bailey and Plouffe). Therefore, no such sums for are known for .

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

	(38)
	(39)
	(40)
	(41)

Here, each subsequent multiplication by the nth prime leaves only terms that are powers of . 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 q and r are the primes congruent to 1 and 3 modulo 4, respectively.

For even ,

where is a Bernoulli number (Mathews and Walker 1964, 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 n.) Rewriting (46),

for n = 1, 3, ... (Havil 2003, p. 194), where is a Bernoulli number, the first few values of which are , 1/120, , 1/240, ... (Sloane's A001067 and A006953).

Although no analytic form for is known for odd n,

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

for n = 3, 5, ... (Apostol 1973, given incorrectly in Stark 1974).

For the Möbius function,

(Havil 2003, p. 209).

The values of for small positive integer values of n are

Euler gave to for even n (Wells 1986, p. 54), and Stieltjes (1993) determined the values of , ..., to 30 digits of accuracy in 1887. The denominators of for n = 1, 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (Sloane's A002432).

An integral for even n is given by

and integrals for odd n are given by

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

The value of can be computed by performing the inner sum in equation (18) with s = 0,

to obtain

where is the Kronecker delta. Similarly, the value of can be computed by performing the inner sum in equation (18) with s = -1,

which gives

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

Rapidly converging series for for n odd were first discovered by Ramanujan (Zucker 1979, Zucker 1984, Berndt 1988, Bailey et al. 1997, Cohen 2000). For n > 1 and ,

where is again a Bernoulli number and is a binomial coefficient. The values of the left-hand sums (divided by ) in (71) for n = 3, 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/390769879500, 7708537/21438612514068750, ... (Sloane's A057866 and A057867). For and , the corresponding formula is slightly messier,

(Cohen 2000).

Defining

the first few values can then be written

(Plouffe).

A number of sum identities involving include

A surprising sum involving is given by

where is the Euler-Mascheroni constant (Havil 2003, pp. 109 and 111-112). Other unexpected sums are

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

(89) is a special case of

(Danese 1967; Boros and Moll 2004, p. 248).

An additional set of sums over is given by

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

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

 

Abel's Functional Equation, Berry Conjecture, Critical Line, Critical Strip, Debye Functions, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, Euler Product, Harmonic Series, Hurwitz Zeta Function, Khinchin's Constant, Lehmer's Phenomenon, Montgomery's Pair Correlation Conjecture, Periodic Zeta Function, Prime Number Theorem, Psi Function, Riemann Hypothesis, Riemann P-Series, Riemann-Siegel Functions, Riemann-von Mangoldt Formula, Riemann Zeta Function Zeta(2), Riemann Zeta Function Zeros, Stieltjes Constants, Voronin Universality Theorem, Xi-Function