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].
The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not a coincidence since it turns out that monotonic decrease implies the Riemann hypothesis (Zvengrowski and Saidak 2003; Borwein and Bailey 2003, pp. 9596).
On the real line with , the Riemann zeta function can be defined by the integral
(1)

where is the gamma function. If is an integer , then we have the identity
(2)
 
(3)
 
(4)

so
(5)

To evaluate , let so that and plug in the above identity to obtain
(6)
 
(7)
 
(8)

Integrating the final expression in (8) gives , which cancels the factor and gives the most common form of the Riemann zeta function,
(9)

which is sometimes known as a pseries.
The Riemann zeta function can also be defined in terms of multiple integrals by
(10)

and as a Mellin transform by
(11)

for , where is the fractional part (Balazard and Saias 2000).
It appears in the unit square integral
(12)

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
(13)

(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
(14)

(Ayoub 1974), which was proved by Riemann for all complex (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 , which corresponds to a simple pole with complex residue 1 (Krantz 1999, p. 160). In particular, as , obeys
(15)

where is the EulerMascheroni constant (Whittaker and Watson 1990, p. 271).
To perform the analytic continuation for , write
(16)
 
(17)
 
(18)

so rewriting in terms of immediately gives
(19)

Therefore,
(20)

Here, the sum on the righthand side is exactly the Dirichlet eta function (sometimes also called the alternating zeta function). While this formula defines for only the right halfplane , equation (◇) 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 ) is given by
(21)

(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 to equation (20).
Hasse (1930) also proved the related globally (but more slowly) convergent series
(22)

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
(23)

(If the singular term is excluded from the sum definition of , then as well.) Expanding about gives
(24)

where are the socalled Stieltjes constants.
The Riemann zeta function can also be defined in the complex plane by the contour integral
(25)

for all , where the contour is illustrated above (Havil 2003, pp. 193 and 249252).
Zeros of come in (at least) two different types. Socalled "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" at certain
(26)

for 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 line) as is varied from 0 to 35 (Derbyshire 2004, p. 221).
The Riemann zeta function can be split up into
(27)

where and are the RiemannSiegel functions.
The Riemann zeta function is related to the Dirichlet lambda function and Dirichlet eta function by
(28)

and
(29)

(Spanier and Oldham 1987).
It is related to the Liouville function by
(30)

(Lehman 1960, Hardy and Wright 1979). Furthermore,
(31)

where is the number of distinct prime factors of (Hardy and Wright 1979, p. 254).
For a positive even integer , , ...,
(32)

giving the first few as
(33)
 
(34)
 
(35)
 
(36)

(OEIS A117972 and A117973). For ,
(37)

where is the GlaisherKinkelin constant. Using equation (◇) gives the derivative
(38)

which can be derived directly from the Wallis formula (Sondow 1994). can also be derived directly from the EulerMaclaurin summation formula (Edwards 2001, pp. 134135). In general, can be expressed analytically in terms of , , the EulerMascheroni constant , and the Stieltjes constants , with the first few examples being
(39)
 
(40)

Derivatives can also be given in closed form, for example,
(41)
 
(42)

(OEIS A114875).
The derivative of the Riemann zeta function for is defined by
(43)
 
(44)

can be given in closed form as
(45)
 
(46)

(OEIS A073002), where is the GlaisherKinkelin constant (given in series form by Glaisher 1894).
The series for about is
(47)

where are Stieltjes constants.
In 1739, Euler found the rational coefficients 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 . 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.
A number of interesting sums for , with a positive integer, can be written in terms of binomial coefficients as the binomial sums
(48)
 
(49)
 
(50)

(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
(51)

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 if algebraic of degree 25 or less, then the norm of coefficients must exceed (Bailey et al. 2007, pp. 7071, updating Bailey and Plouffe). Therefore, no such sums for are known for .
The identity
(52)
 
(53)
 
(54)
 
(55)

for is complex number not equal to a nonzero integer gives an Apérylike formula for even positive (Bailey et al. 2006, pp. 7277).
The Riemann zeta function may be computed analytically for even 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,
(56)
 
(57)
 
(58)
 
(59)
 
(60)

Here, each subsequent multiplication by the th prime leaves only terms that are powers of . Therefore,
(61)

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. 104106). The formula can also be written
(62)

where and are the primes congruent to 1 and 3 modulo 4, respectively.
For even ,
(63)

where is a Bernoulli number (Mathews and Walker 1970, pp. 5053; Havil 2003, p. 194). Another intimate connection with the Bernoulli numbers is provided by
(64)

for , which can be written
(65)

for . (In both cases, only the even cases are of interest since trivially for odd .) Rewriting (65),
(66)

for , 3, ... (Havil 2003, p. 194), where is a Bernoulli number, the first few values of which are , 1/120, , 1/240, ... (OEIS A001067 and A006953).
Although no analytic form for is known for odd ,
(67)

where is a harmonic number (Stark 1974). In addition, can be expressed as the sum limit
(68)

for , 5, ... (Apostol 1973, given incorrectly in Stark 1974).
For the Möbius function,
(69)

(Havil 2003, p. 209).
The values of for small positive integer values of are
(70)
 
(71)
 
(72)
 
(73)
 
(74)
 
(75)
 
(76)
 
(77)
 
(78)
 
(79)

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 for , 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (OEIS A002432). The numbers of decimal digits in the denominators of for , 1, ... are 1, 5, 133, 2277, 32660, 426486, 5264705, ... (OEIS A114474).
An integral for positive even integers is given by
(80)

and integrals for positive odd integers are given by
(81)
 
(82)
 
(83)
 
(84)

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 (◇) with ,
(85)

to obtain
(86)

where is the Kronecker delta.
Similarly, the value of can be computed by performing the inner sum in equation (◇) with ,
(87)

which gives
(88)
 
(89)
 
(90)

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
(91)

(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 ,
(92)

where is again a Bernoulli number and is a binomial coefficient. The values of the lefthand sums (divided by ) in (92) 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,
(93)

(Cohen 2000).
Defining
(94)

the first few values can then be written
(95)
 
(96)
 
(97)
 
(98)
 
(99)
 
(100)
 
(101)
 
(102)
 
(103)
 
(104)

(Plouffe 1998).
Another set of related formulas are
(105)
 
(106)
 
(107)
 
(108)
 
(109)

(Plouffe 2006).
Multiterm sums for odd include
(110)
 
(111)
 
(112)
 
(113)

(Borwein and Bradley 1996, 1997; Bailey et al. 2007, p. 71), where is a generalized harmonic number.
G. Huvent (2002) found the beautiful formula
(114)

A number of sum identities involving include
(115)
 
(116)
 
(117)
 
(118)

Sums involving integers multiples of the argument include
(119)
 
(120)
 
(121)

where is a harmonic number.
Two surprising sums involving are given by
(122)
 
(123)

where is the EulerMascheroni constant (Havil 2003, pp. 109 and 111112). Equation (122) can be generalized to
(124)

(T. Drane, pers. comm., Jul. 7, 2006) for .
Other unexpected sums are
(125)

(Tyler and Chernhoff 1985; Boros and Moll 2004, p. 248) and
(126)

(125) is a special case of
(127)

where is a Hurwitz zeta function (Danese 1967; Boros and Moll 2004, p. 248).
Considering the sum
(128)

then
(129)

where is the natural logarithm of 2, which is a particular case of
(130)

where is the digamma function and is the EulerMascheroni constant, which can be derived from
(131)

(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
(132)

where is a binomial coefficient (B. Cloitre, pers. comm., Sep. 20, 2005).
An additional set of sums over is given by
(133)
 
(134)
 
(135)
 
(136)
 
(137)
 
(138)
 
(139)
 
(140)
 
(141)
 
(142)
 
(143)

(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 closedform expression.
The inverse of the Riemann zeta function , plotted above, is the asymptotic density of thpowerfree numbers (i.e., squarefree numbers, cubefree numbers, etc.). The following table gives the number of thpowerfree numbers for several values of .
2  0.607927  7  61  608  6083  60794  607926 
3  0.831907  9  85  833  8319  83190  831910 
4  0.923938  10  93  925  9240  92395  923939 
5  0.964387  10  97  965  9645  96440  964388 
6  0.982953  10  99  984  9831  98297  982954 