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Riemann Zeta Function


RiemannZeta

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 zeta(s) and is plotted above (using two different scales) along the real axis.

RiemannZetaReImAbs
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In general, zeta(s) is defined over the complex plane for one complex variable, which 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).

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

RiemannZetaRidges

The plot above shows the "ridges" of |zeta(x+iy)| for 0<x<1 and 1<y<100. The fact that the ridges appear to decrease monotonically for 0<=x<=1/2 is not a coincidence since it turns out that monotonic decrease implies the Riemann hypothesis (Zvengrowski and Saidak 2003; Borwein and Bailey 2003, pp. 95-96).

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

 zeta(x)=1/(Gamma(x))int_0^infty(u^(x-1))/(e^u-1)du,
(1)

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

(u^(n-1))/(e^u-1)=(e^(-u)u^(n-1))/(1-e^(-u))
(2)
=e^(-u)u^(n-1)sum_(k=0)^(infty)e^(-ku)
(3)
=sum_(k=1)^(infty)e^(-ku)u^(n-1),
(4)

so

 int_0^infty(u^(n-1))/(e^u-1)du=sum_(k=1)^inftyint_0^inftye^(-ku)u^(n-1)du.
(5)

To evaluate zeta(n), let y=ku so that dy=kdu and plug in the above identity to obtain

zeta(n)=1/(Gamma(n))sum_(k=1)^(infty)int_0^inftye^(-ku)u^(n-1)du
(6)
=1/(Gamma(n))sum_(k=1)^(infty)int_0^inftye^(-y)(y/k)^(n-1)(dy)/k
(7)
=1/(Gamma(n))sum_(k=1)^(infty)1/(k^n)int_0^inftye^(-y)y^(n-1)dy.
(8)

Integrating the final expression in (8) gives Gamma(n), which cancels the factor 1/Gamma(n) and gives the most common form of the Riemann zeta function,

 zeta(n)=sum_(k=1)^infty1/(k^n),
(9)

which is sometimes known as a p-series.

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

 zeta(n)=int_0^1...int_0^1_()_(n)(product_(i=1)^(n)dx_i)/(1-product_(i=1)^(n)x_i),
(10)

and as a Mellin transform by

 int_0^inftyfrac(1/t)t^(s-1)dt=-(zeta(s))/s
(11)

for 0<R[s]<1, where frac(x) is the fractional part (Balazard and Saias 2000).

It appears in the unit square integral

 int_0^1int_0^1([-ln(xy)]^s)/(1-xy)dxdy=Gamma(s+2)zeta(s+2),
(12)

valid for R[s]>1 (Guillera and Sondow 2005). For s a nonnegative integer, this formula is due to Hadjicostas (2002), and the special cases s=0 and s=1 are due to Beukers (1979).

Note that the zeta function zeta(s) has a singularity at s=1, where it reduces to the divergent harmonic series.

The Riemann zeta function satisfies the reflection functional equation

 zeta(1-s)=2(2pi)^(-s)cos(1/2spi)Gamma(s)zeta(s)
(13)

(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

 Gamma(s/2)pi^(-s/2)zeta(s)=Gamma((1-s)/2)pi^(-(1-s)/2)zeta(1-s)
(14)

(Ayoub 1974), which was proved by Riemann for all complex s (Riemann 1859).

As defined above, the zeta function zeta(s) with s=sigma+it a complex number is defined for R[s]>1. However, zeta(s) 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 s->1, zeta(s) obeys

 lim_(s->1)[zeta(s)-1/(s-1)]=gamma,
(15)

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

To perform the analytic continuation for R[s]>0, write

sum_(n=1)^(infty)((-1)^n)/(n^s)+sum_(n=1)^(infty)1/(n^s)=2sum_(n=2,4,...)^(infty)1/(n^s)
(16)
=2sum_(k=1)^(infty)1/((2k)^s)
(17)
=2^(1-s)sum_(k=1)^(infty)1/(k^s),
(18)

so rewriting in terms of zeta(s) immediately gives

 sum_(n=1)^infty((-1)^n)/(n^s)+zeta(s)=2^(1-s)zeta(s).
(19)

Therefore,

 zeta(s)=1/(1-2^(1-s))sum_(n=1)^infty((-1)^(n-1))/(n^s).
(20)

Here, the sum on the right-hand side is exactly the Dirichlet eta function eta(s) (sometimes also called the alternating zeta function). While this formula defines zeta(s) for only the right half-plane R[s]>0, 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 zeta(s) to the entire complex plane except s=1) is given by

 zeta(s)=1/(1-2^(1-s))sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k)(k+1)^(-s)
(21)

(Havil 2003, p. 206), where (n; k) 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 (20).

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

 zeta(s)=1/(s-1)sum_(n=0)^infty1/(n+1)sum_(k=0)^n(-1)^k(n; k)(k+1)^(1-s)
(22)

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

 zeta(s)=zeta(s,1).
(23)

(If the singular term is excluded from the sum definition of zeta(s,a), then zeta(s)=zeta(s,0) as well.) Expanding zeta(s) about s=1 gives

 zeta(s)=1/(s-1)+sum_(n=0)^infty((-1)^n)/(n!)gamma_n(s-1)^n,
(24)

where gamma_n are the so-called Stieltjes constants.

RiemannZetaFunctionGamma

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

 zeta(z)=(Gamma(1-z))/(2pii)∮_gamma(u^(z-1))/(e^(-u)-1)du
(25)

for all z!=1, where the contour is illustrated above (Havil 2003, pp. 193 and 249-252).

Zeros of zeta(s) 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

 s=sigma+it
(26)

for s in the "critical strip" 0<sigma<1. The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of zeta(s) all have real part sigma=R[s]=1/2, a line called the "critical line." This is now known to be true for the first 250×10^9 roots.

RiemannZetaCriticalStrip

The plot above shows the real and imaginary parts of zeta(1/2+iy) (i.e., values of zeta(z) along the critical line) as y is varied from 0 to 35 (Derbyshire 2004, p. 221).

The Riemann zeta function can be split up into

 zeta(1/2+it)=Z(t)e^(-itheta(t)),
(27)

where Z(t) and theta(t) are the Riemann-Siegel functions.

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

 (zeta(nu))/(2^nu)=(lambda(nu))/(2^nu-1)=(eta(nu))/(2^nu-2)
(28)

and

 zeta(nu)+eta(nu)=2lambda(nu)
(29)

(Spanier and Oldham 1987).

It is related to the Liouville function lambda(n) by

 (zeta(2s))/(zeta(s))=sum_(n=1)^infty(lambda(n))/(n^s)
(30)

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

 (zeta^2(s))/(zeta(2s))=sum_(n=1)^infty(2^(omega(n)))/(n^s),
(31)

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

For -2n a positive even integer -2, -4, ...,

 zeta^'(-2n)=((-1)^nzeta(2n+1)(2n)!)/(2^(2n+1)pi^(2n)),
(32)

giving the first few as

zeta^'(-2)=-(zeta(3))/(4pi^2)
(33)
zeta^'(-4)=(3zeta(5))/(4pi^4)
(34)
zeta^'(-6)=-(45zeta(7))/(8pi^6)
(35)
zeta^'(-8)=(315zeta(9))/(4pi^8)
(36)

(OEIS A117972 and A117973). For n=-1,

 zeta^'(-1)=1/(12)-lnA,
(37)

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

 zeta^'(0)=-1/2ln(2pi),
(38)

which can be derived directly from the Wallis formula (Sondow 1994). zeta^'(0)/zeta(0)=ln(2pi) can also be derived directly from the Euler-Maclaurin summation formula (Edwards 2001, pp. 134-135). In general, zeta^((n))(0) can be expressed analytically in terms of pi, zeta(n), the Euler-Mascheroni constant gamma, and the Stieltjes constants gamma_i, with the first few examples being

zeta^('')(0)=gamma_1+1/2gamma^2-1/(24)pi^2-1/2[ln(2pi)]^2
(39)
zeta^(''')(0)=3ln(2pi)gamma_1+3gammagamma_1+3/2gamma_2-zeta(3)-1/2[ln(2pi)]^3-1/8pi^2ln(2pi)+3/2gamma^2ln(2pi)+gamma^3.
(40)

Derivatives zeta^((n))(1/2) can also be given in closed form, for example,

zeta^'(1/2)=1/4[(pi+2gamma+6ln2+2lnpi)zeta(1/2)]
(41)
=-3.92264613...
(42)

(OEIS A114875).

The derivative of the Riemann zeta function for R[s]>1 is defined by

zeta^'(s)=-sum_(k=1)^(infty)(lnk)/(k^s)
(43)
=-sum_(k=2)^(infty)(lnk)/(k^s).
(44)

zeta^'(2) can be given in closed form as

zeta^'(2)=1/6pi^2[gamma+ln(2pi)-12lnA]
(45)
=-0.93754825431...
(46)

(OEIS A073002), where A is the Glaisher-Kinkelin constant (given in series form by Glaisher 1894).

The series for zeta^'(s) about s=1 is

 zeta^'(s)=-1/((s-1)^2)-gamma_1+gamma_2(s-1)-1/2gamma_3(s-1)^2+...,
(47)

where gamma_i are Stieltjes constants.

In 1739, Euler found the rational coefficients C in zeta(2n)=Cpi^(2n) in terms of the Bernoulli numbers. Which, when combined with the 1882 proof by Lindemann that pi is transcendental, effectively proves that zeta(2n) is transcendental. The study of zeta(2n+1) is significantly more difficult. Apéry (1979) finally proved zeta(3) to be irrational, but no similar results are known for other odd n. As a result of Apéry's important discovery, zeta(3) is sometimes called Apéry's constant. Rivoal (2000) and Ball and Rivoal (2001) proved that there are infinitely many integers n such that zeta(2n+1) is irrational, and subsequently that at least one of zeta(5), zeta(7), ..., zeta(21) is irrational (Rivoal 2001). This result was subsequently tightened by Zudilin (2001), who showed that at least one of zeta(5), zeta(7), zeta(9), or zeta(11) is irrational.

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

zeta(2)=3sum_(k=1)^(infty)1/(k^2(2k; k))
(48)
zeta(3)=5/2sum_(k=1)^(infty)((-1)^(k-1))/(k^3(2k; k))
(49)
zeta(4)=(36)/(17)sum_(k=1)^(infty)1/(k^4(2k; k))
(50)

(Guy 1994, p. 257; Bailey et al. 2007, p. 70). Apéry arrived at his result with the aid of the k^(-3) sum formula above. A relation of the form

 zeta(5)=Z_5sum_(k=1)^infty((-1)^(k-1))/(k^5(2k; k))
(51)

has been searched for with Z_5 a rational or algebraic number, but if Z_5 is a root of a polynomial of degree 25 or less, then the Euclidean norm of the coefficients must be larger than 1.24×10^(383), and if zeta(5) if algebraic of degree 25 or less, then the norm of coefficients must exceed 1.98×10^(380) (Bailey et al. 2007, pp. 70-71, updating Bailey and Plouffe). Therefore, no such sums for zeta(n) are known for n>=5.

The identity

sum_(k=1)^(infty)1/(k^2-x^2)=sum_(n=0)^(infty)zeta(2n+2)x^(2n)
(52)
=(1-pixcot(pix))/(2x^2)
(53)
=3sum_(k=1)^(infty)1/(k^2(2k; k)(1-(x^2)/(k^2)))product_(m=1)^(k-1)(1-(4x^2)/(m^2))/(1-(x^2)/(m^2))
(54)
=(3_4F_3(1,2,1-2x,1+2x;3/2,2-x,2+x;1/4))/(2(1-x^2))
(55)

for x is complex number not equal to a nonzero integer gives an Apéry-like formula for even positive n (Bailey et al. 2006, pp. 72-77).

The Riemann zeta function zeta(2n) 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 a product over the primes was first discovered by Euler in 1737,

zeta(s)(1-2^(-s))=(1+1/(2^s)+1/(3^s)+...)(1-1/(2^s))
(56)
=(1+1/(2^s)+1/(3^s)+...)-(1/(2^s)+1/(4^s)+1/(6^s)+...)
(57)
zeta(s)(1-2^(-s))(1-3^(-s))=(1+1/(3^s)+1/(5^s)+1/(7^s)+...)-(1/(3^s)+1/(9^s)+1/(15^s)+...)
(58)
zeta(s)(1-2^(-s))(1-3^(-s))...(1-p_n^(-s))...=zeta(s)product_(n=1)^(infty)(1-p_n^(-s))
(59)
=1.
(60)

Here, each subsequent multiplication by the nth prime p_n leaves only terms that are powers of p^(-s). Therefore,

 zeta(s)=[product_(n=1)^infty(1-p_n^(-s))]^(-1),
(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. 104-106). The formula can also be written

 zeta(s)=(1-2^(-s))^(-1)product_(q=1; (mod 4))(1-q^(-s))^(-1)product_(r=3; (mod 4))(1-r^(-s))^(-1),
(62)

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

For even n>=2,

 zeta(n)=(2^(n-1)|B_n|pi^n)/(n!),
(63)

where B_n 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

 B_n=(-1)^(n+1)nzeta(1-n)
(64)

for n>=1, which can be written

 B_n=-nzeta(1-n)
(65)

for n>=2. (In both cases, only the even cases are of interest since B_n=0 trivially for odd n.) Rewriting (65),

 zeta(-n)=-(B_(n+1))/(n+1)
(66)

for n=1, 3, ... (Havil 2003, p. 194), where B_n is a Bernoulli number, the first few values of which are -1/12, 1/120, -1/252, 1/240, ... (OEIS A001067 and A006953).

Although no analytic form for zeta(n) is known for odd n,

 zeta(3)=1/2sum_(k=1)^infty(H_k)/(k^2),
(67)

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

 zeta(n)=lim_(x->infty)1/((2x+1)^n)sum_(k=1)^x[cot(k/(2x+1))]^n
(68)

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

For mu(n) the Möbius function,

 1/(zeta(s))=sum_(n=1)^infty(mu(n))/(n^s)
(69)

(Havil 2003, p. 209).

The values of zeta(n) for small positive integer values of n are

zeta(1)=infty
(70)
zeta(2)=(pi^2)/6
(71)
zeta(3)=1.2020569032...
(72)
zeta(4)=(pi^4)/(90)
(73)
zeta(5)=1.0369277551...
(74)
zeta(6)=(pi^6)/(945)
(75)
zeta(7)=1.0083492774...
(76)
zeta(8)=(pi^8)/(9450)
(77)
zeta(9)=1.0020083928...
(78)
zeta(10)=(pi^(10))/(93555).
(79)

Euler gave zeta(2) to zeta(26) for even n (Wells 1986, p. 54), and Stieltjes (1993) determined the values of zeta(2), ..., zeta(70) to 30 digits of accuracy in 1887. The denominators of zeta(2n) for n=1, 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (OEIS A002432). The numbers of decimal digits in the denominators of zeta(10^n) for n=0, 1, ... are 1, 5, 133, 2277, 32660, 426486, 5264705, ... (OEIS A114474).

An integral for positive even integers is given by

 zeta(2n)=((-1)^(n+1)2^(2n-3)pi^(2n))/((2^(2n)-1)(2n-2)!)int_0^1E_(2(n-1))(x)dx,
(80)

and integrals for positive odd integers are given by

zeta(2n+1)=((-1)^n2^(2n-1)pi^(2n+1))/((2^(2n+1)-1)(2n)!)int_0^1E_(2n)(x)tan(1/2pix)dx
(81)
=((-1)^n2^(2n-1)pi^(2n+1))/((2^(2n+1)-1)(2n)!)int_0^1E_(2n)(x)cot(1/2pix)dx
(82)
=((-1)^n2^(2n)pi^(2n+1))/((2n+1)!)int_0^1B_(2n+1)(x)tan(1/2pix)dx
(83)
=((-1)^(n+1)2^(2n)pi^(2n+1))/((2n+1)!)int_0^1B_(2n+1)(x)cot(1/2pix)dx,
(84)

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

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

 zeta(0)=-sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k),
(85)

to obtain

 zeta(0)=-sum_(n=0)^infty(delta_(0,n))/(2^(n+1))=-1/(2^(0+1))=-1/2,
(86)

where delta_(0,n) is the Kronecker delta.

Similarly, the value of zeta(-1) can be computed by performing the inner sum in equation (◇) with s=-1,

 zeta(-1)=-1/3sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k)(k+1),
(87)

which gives

zeta(-1)=-1/3sum_(n=0)^(infty)(delta_(0,n)-ndelta_(1,n))/(2^(n+1))
(88)
=-1/3(1/(2^(0+1))-1/(2^(1+1)))
(89)
=-1/(12).
(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

 zeta(1/2)=-1.46035450880...
(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 zeta(n) for n odd were first discovered by Ramanujan (Zucker 1979, 1984, Berndt 1988, Bailey et al. 1997, Cohen 2000). For n>1 and n=3 (mod 4),

 zeta(n)=(2^(n-1)pi^n)/((n+1)!)sum_(k=0)^((n+1)/2)(-1)^(k-1)(n+1; 2k)B_(n+1-2k)B_(2k)-2sum_(k=1)^infty1/(k^n(e^(2pik)-1)),
(92)

where B_k is again a Bernoulli number and (n; k) is a binomial coefficient. The values of the left-hand sums (divided by pi^n) in (92) for n=3, 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/390769879500, 7708537/21438612514068750, ... (OEIS A057866 and A057867). For n>=5 and n=1 (mod 4), the corresponding formula is slightly messier,

 zeta(n)=((2pi)^n)/((n+1)!(n-1))sum_(k=0)^((n+1)/4)(-1)^k(n+1-4k)(n+1; 2k)B_(n+1-2k)B_(2k)-2sum_(k=1)^infty(e^(2pik)(1+(4pik)/(n-1))-1)/(k^n(e^(2pik)-1)^2)
(93)

(Cohen 2000).

Defining

 S_+/-(n)=sum_(k=1)^infty1/(k^n(e^(2pik)+/-1)),
(94)

the first few values can then be written

zeta(3)=7/(180)pi^3-2S_-(3)
(95)
zeta(5)=1/(294)pi^5-(72)/(35)S_-(5)-2/(35)S_+(5)
(96)
zeta(7)=(19)/(56700)pi^7-2S_-(7)
(97)
zeta(9)=(125)/(3704778)pi^9-(992)/(495)S_-(9)-2/(495)S_+(9)
(98)
zeta(11)=(1453)/(425675250)pi^(11)-2S_-(11)
(99)
zeta(13)=(89)/(257432175)pi^(13)-(16512)/(8255)S_-(13)-2/(8255)S_+(13)
(100)
zeta(15)=(13687)/(390769879500)pi^(15)-2S_-(15)
(101)
zeta(17)=(397549)/(112024529867250)pi^(17)-(261632)/(130815)S_-(17)-2/(130815)S_+(17)
(102)
zeta(19)=(7708537)/(21438612514068750)pi^(19)-2S_-(19)
(103)
zeta(21)=(68529640373)/(1881063815762259253125)pi^(21)-(4196352)/(2098175)S_-(21)-2/(2098175)S_+(21)
(104)

(Plouffe 1998).

Another set of related formulas are

zeta(3)=(pi^3)/(28)+(16)/7sum_(n=1)^(infty)1/(n^3(e^(npi)+1))-2/7sum_(n=1)^(infty)1/(n^3(e^(2pin)+1))
(105)
zeta(5)=24sum_(n=1)^(infty)1/(n^5(e^(npi)-1))-(259)/(10)sum_(n=1)^(infty)1/(n^5(e^(2pin)-1))-1/(10)sum_(n=1)^(infty)1/(n^5(e^(4pin)-1))
(106)
zeta(5)=-(7pi^5)/(1840)+(328)/(115)sum_(n=1)^(infty)1/(n^5(e^(pin)-1))-(419)/(460)sum_(n=1)^(infty)1/(n^5(e^(2pin)-1))-9/(115)sum_(n=1)^(infty)1/(n^5(e^(3pin)-1))+(261)/(1840)sum_(n=1)^(infty)1/(n^5(e^(6pin)-1))-9/(1840)sum_(n=1)^(infty)1/(n^5(e^(12pin)-1))
(107)
zeta(7)=(304)/(13)sum_(n=1)^(infty)1/(n^7(e^(pin)-1))-(103)/4sum_(n=1)^(infty)1/(n^7(e^(2pin)-1))-(19)/(52)sum_(n=1)^(infty)1/(n^7(e^(4pin)-1))
(108)
zeta(9)=(64)/3sum_(n=1)^(infty)1/(n^9(e^(pin)-1))+(441)/(20)sum_(n=1)^(infty)1/(n^9(e^(2pin)-1))-32sum_(n=1)^(infty)1/(n^9(e^(3pin)-1))-(4763)/(60)sum_(n=1)^(infty)1/(n^9(e^(4pin)-1))+(529)/8sum_(n=1)^(infty)1/(n^9(e^(6pin)-1))-1/8sum_(n=1)^(infty)1/(n^9(e^(12pin)-1))
(109)

(Plouffe 2006).

Multiterm sums for odd zeta(n) include

zeta(5)=2sum_(k=1)^(infty)((-1)^(k+1))/(k^5(2k; k))-5/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2)))/(k^3(2k; k))
(110)
zeta(7)=5/2sum_(k=1)^(infty)((-1)^(k+1))/(k^7(2k; k))+(25)/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^3(2k; k))
(111)
zeta(9)=9/4sum_(k=1)^(infty)((-1)^(k+1))/(k^9(2k; k))-5/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2)))/(k^7(2k; k))+5sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^5(2k; k))+(45)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((6)))/(k^3(2k; k))-(25)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2))H_(k-1)^((4)))/(k^3(2k; k))
(112)
zeta(11)=5/2sum_(k=1)^(infty)((-1)^(k+1))/(k^(11)(2k; k))+(25)/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^7(2k; k))-(75)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((8)))/(k^3(2k; k))+(125)/4sum_(k=1)^(infty)((-1)^(k+1)[H_(k-1)^((4))]^2)/(k^3(2k; k))
(113)

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

G. Huvent (2002) found the beautiful formula

 zeta(5)=-(16)/(11)sum_(n=1)^infty([2(-1)^n+1]H_n)/(n^4).
(114)

A number of sum identities involving zeta(n) include

sum_(n=2)^(infty)[zeta(n)-1]=1
(115)
sum_(n=2,4,...)^(infty)[zeta(n)-1]=3/4
(116)
sum_(n=3,5,...)^(infty)[zeta(n)-1]=1/4
(117)
sum_(n=2)^(infty)(-1)^n[zeta(n)-1]=1/2.
(118)

Sums involving integers multiples of the argument include

sum_(n=1)^(infty)[zeta(2n)-1]=3/4
(119)
sum_(n=1)^(infty)[zeta(3n)-1]=1/3[-(-1)^(2/3)H_((3-isqrt(3))/2)+(-1)^(1/3)H_((3+isqrt(3))/2)]
(120)
sum_(n=1)^(infty)[zeta(4n)-1]=1/8(7-2cothpi),
(121)

where H_n is a harmonic number.

Two surprising sums involving zeta(x) are given by

sum_(k=2)^(infty)((-1)^kzeta(k))/k=gamma
(122)
sum_(k=2)^(infty)(zeta(k)-1)/k=1-gamma,
(123)

where gamma is the Euler-Mascheroni constant (Havil 2003, pp. 109 and 111-112). Equation (122) can be generalized to

 sum_(k=2)^infty((-x)^kzeta(k))/k=xgamma+ln(x!)
(124)

(T. Drane, pers. comm., Jul. 7, 2006) for -1<x<=1.

Other unexpected sums are

 sum_(n=1)^infty(zeta(2n))/(n(2n+1)2^(2n))=lnpi-1
(125)

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

 sum_(n=1)^infty(zeta(2n))/(n(2n+1))=ln(2pi)-1.
(126)

(125) is a special case of

 sum_(k=1)^infty(zeta(2k,z))/(k(2k+1)2^(2k)) 
 =(2z-1)ln(z-1/2)-2z+1+ln(2pi)-2lnGamma(z),
(127)

where zeta(s,a) is a Hurwitz zeta function (Danese 1967; Boros and Moll 2004, p. 248).

Considering the sum

 S_n=sum_(k=2)^(n-2)(zeta(k)zeta(n-k))/(2^k),
(128)

then

 lim_(n->infty)S_n=ln2,
(129)

where ln2 is the natural logarithm of 2, which is a particular case of

 lim_(n->infty)sum_(k=2)^(n-2)zeta(k)zeta(n-k)x^(k-1)=x^(-1)-psi_0(-x)-gamma,
(130)

where psi_0(z) is the digamma function and gamma is the Euler-Mascheroni constant, which can be derived from

 sum_(k=2)^inftyzeta(k)x^(k-1)=-psi_0(1-x)-gamma
(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 m=1 case) is given by

 sum_(k=1)^infty1/([k(k+1)]^(2m+1))=-2sum_(k=0)^mzeta(2k)(4m-2k+1; 2m),
(132)

where (n; k) is a binomial coefficient (B. Cloitre, pers. comm., Sep. 20, 2005).

An additional set of sums over zeta(n) is given by

C_1=sum_(n=2)^(infty)(zeta(n))/(n!)
(133)
=int_0^infty(I_1(2sqrt(u))-sqrt(u))/((e^u-1)sqrt(u))du
(134)
=int_0^infty(_0F^~_1(;2;u)-1)/(e^u-1)du
(135)
 approx 1.078189
(136)
C_2=sum_(n=1)^(infty)(zeta(2n))/(n!)
(137)
=sum_(n=1)^(infty)e^(1/n^2)-1
(138)
=int_0^1(u_0F_2(;3/2,2;1/4u^4))/(e^u-1)du
(139)
 approx 2.407447
(140)
C_3=sum_(n=1)^(infty)(zeta(2n))/((2n)!)
(141)
=int_0^infty(u+_0F^~_1(;2;-u)-_0F^~_1(;2;u))/(2(1-e^u))du
(142)
 approx 0.869002.
(143)

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

RiemannZetaInv

The inverse of the Riemann zeta function 1/zeta(p), plotted above, is the asymptotic density of pth-powerfree numbers (i.e., squarefree numbers, cubefree numbers, etc.). The following table gives the number Q_p(n) of pth-powerfree numbers <=n for several values of n.

p1/zeta(p)Q_p(10)Q_p(100)Q_p(10^3)Q_p(10^4)Q_p(10^5)Q_p(10^6)
20.607927761608608360794607926
30.831907985833831983190831910
40.9239381093925924092395923939
50.9643871097965964596440964388
60.9829531099984983198297982954

See also

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, p-Series, 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 Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/

Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Riemann Zeta Function and Other Sums of Reciprocal Powers." §23.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.Adamchik, V. S. and Srivastava, H. M. "Some Series of the Zeta and Related Functions." Analysis 18, 131-144, 1998. Aizenberg, L.; Adamchik, V.; and Levit, V. E. "Approaching the Riemann Hypothesis with Mathematica." http://library.wolfram.com/infocenter/Articles/3268/.Apéry, R. "Irrationalité de zeta(2) et zeta(3)." Astérisque 61, 11-13, 1979.Apostol, T. M. "Another Elementary Proof of Euler's Formula for zeta(2n)." Amer. Math. Monthly 80, 425-431, 1973.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 332-335, 1985.Ayoub, R. "Euler and the Zeta Function." Amer. Math. Monthly 81, 1067-1086, 1974.Bailey, D. H. "Multiprecision Translation and Execution of Fortran Programs." ACM Trans. Math. Software. To appear.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khinchin Constant." Math. Comput. 66, 417-431, 1997.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." http://www.cecm.sfu.ca/organics/papers/bailey/.Balazard, M. and Saias, E. "The Nyman-Beurling Equivalent Form for the Riemann Hypothesis." Expos. Math. 18, 131-138, 2000.Balazard, M.; Saias, E.; and Yor, M. "Notes sur la fonction zeta de Riemann, 2." Adv. Math. 143, 284-287, 1999.Ball, K. and Rivoal, T. "Irrationalité d'une infinité valeurs de la fonction zêta aux entiers impairs." Invent. Math. 146, 193-207, 2001.Berndt, B. C. Ch. 14 in Ramanujan's Notebooks, Part II. New York: Springer-Verlag, 1988.Beukers, F. "A Note on the Irrationality of zeta(2) and zeta(3)." Bull. London Math. Soc. 11, 268-272, 1979.Biane, P.; Pitman, J.; and Yor, M. "Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions." Bull. Amer. Math. Soc. 38, 435-465, 2001.Bloch, S. "Zeta Values and Differential Operators on the Circle." J. Algebra 182, 476-500, 1996.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 95-96 and 99-100, 2003.Borwein, D. and Borwein, J. "On an Intriguing Integral and Some Series Related to zeta(4)." Proc. Amer. Math. Soc. 123, 1191-1198, 1995.Borwein, J. M. and Bradley, D. M. "Searching Symbolically for Apéry-Like Formulae for Values of the Riemann Zeta Function." ACM SIGSAM Bull. Algebraic Sym. Manip. 30, 2-7, 1996.Borwein, J. M. and Bradley, D. M. "Empirically Determined Apéry-Like Formulae for zeta(4n+3)." Exp. Math. 6, 181-194, 1997.Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math. 121, 247-296, 2000.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Choudhury, B. K. "The Riemann Zeta-Function and Its Derivatives." Proc. Roy. Soc. London Ser. A 450, 477-499, 1995.Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi. 2000.Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. http://www.ams.org/notices/200303/fea-conrey-web.pdf.Cvijović, D. and Klinowski, J. "Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments." J. Comput. Appl. Math. 142, 435-439, 2002.Danese, A. E. "Solution to Problem 1801. A Zeta-Function Identity." Amer. Math. Monthly 74, 80-81, 1967.Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Elizalde, E. Ten Physical Applications of Spectral Zeta Functions. Berlin: Springer-Verlag, 1995.Elizalde, E.; Odintsov, S. D.; Romeo, A.; Bytsenko, A. A.; and Zerbini, S. Zeta Regularization Techniques With Applications. River Edge, NJ: World Scientific, 1994.Euler, L. "Remarques sur un beau rapport entre les series des puissances tant directes que réciproques." Mémoires de l'academie des sciences de Berlin 17, 83-106, 1768. Reprinted in Opera Omnia, Series 1, Vol. 15 pp. 70-90.Glaisher, J. W. L. "On the Constant which Occurs in the Formula for 1^1.2^2.3^3...n^n." Messenger Math. 24, 1-16, 1894.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.Guy, R. K. "Series Associated with the zeta-Function." §F17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 257-258, 1994.Hadjicostas, P. "Some Generalizations of Beukers' Integrals." Kyungpook Math. J. 42, 399-416, 2002.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Wright, E. M. "The Zeta Function." §17.2 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 245-247 and 255, 1979.Hasse, H. "Ein Summierungsverfahren für die Riemannsche Zeta-Reihe." Math. Z. 32, 458-464, 1930.Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Howson, A. G. "Addendum to: 'Euler and the Zeta Function' (Amer. Math. Monthly 81 (1974), 1067-1086) by Raymond Ayoub." Amer. Math. Monthly 82, 737, 1975.Huvent, G. "Autour de la primitive de t^pcoth(alphat/2)." Feb. 3, 2002. http://perso.orange.fr/gery.huvent/articlespdf/Autour_primitive.pdf.Huylebrouck, D. "Similarities in Irrationality Proofs for pi, ln2, zeta(2), and zeta(3)." Amer. Math. Monthly 108, 222-231, 2001.Ivic, A. A. The Riemann Zeta-Function. New York: Wiley, 1985.Ivic, A. A. Lectures on Mean Values of the Riemann Zeta Function. Berlin: Springer-Verlag, 1991.Jones, G. A. and Jones, J. M. "The Riemann Zeta Function." Ch. 9 in Elementary Number Theory. Berlin: Springer-Verlag, pp. 163-189, 1998.Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta-Function. Hawthorne, NY: De Gruyter, 1992.Katayama, K. "On Ramanujan's Formula for Values of Riemann Zeta-Function at Positive Odd Integers." Acta Math. 22, 149-155, 1973.Keiper, J. "The Zeta Function of Riemann." Mathematica Educ. Res. 4, 5-7, 1995.Knopp, K. "4th Example: The Riemann zeta-Function." Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 51-57, 1996.Krantz, S. G. "Riemann's Zeta Function." §13.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 158-159, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 35, 1983.Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.Lepowski, J. "Vertex Operator Algebras and the Zeta Function." 29 Sep 1999. http://arxiv.org/abs/math.QA/9909178.Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison-Wesley, pp. 50-53, 1970.Patterson, S. J. An Introduction to the Theory of the Riemann Zeta-Function. New York: Cambridge University Press, 1988.Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. http://mathworld.wolfram.com/news/2004-11-08/seventidbits/#3.Plouffe, S. "Identities Inspired from Ramanujan Notebooks II." Jul. 21, 1998. http://www.lacim.uqam.ca/~plouffe/identities.html.Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859. Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972.Rivoal, T. "La fonction Zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs." Comptes Rendus Acad. Sci. Paris 331, 267-270, 2000.Rivoal, T. "Irrationalité d'au moins un des neuf nombres zeta(5), zeta(7), ..., zeta(21)." 25 Apr 2001. http://arxiv.org/abs/math.NT/0104221.Sloane, N. J. A. Sequences A001067, A002432/M4283, A006953/M2039, A057866, A057867, A059750, A073002, A076813, A093720, A093721, A114474, A114875, A117972, and A117973 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." Proc. Amer. Math. Soc. 120, 421-424, 1994.Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." J. Math. Anal. Appl. 246, 331-351, 2000.Stark, E. L. "The Series sum_(k=1)^(infty)k^(-s) s=2, 3, 4, ..., Once More." Math. Mag. 47, 197-202, 1974.Stieltjes, T. J. Oeuvres Complètes, Vol. 2 (Ed. G. van Dijk.) New York: Springer-Verlag, p. 100, 1993.Titchmarsh, E. C. The Zeta-Function of Riemann. London: Cambridge University Press, 1930.Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.Tyler, D. and Chernhoff, P. "Problem 3103. An Odd Sum Reappears." Amer. Math. Monthly 92, 507, 1985.Vardi, I. "The Riemann Zeta Function." Ch. 8 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 141-174, 1991.Wagon, S. "The Riemann Zeta Function." §10.6 in Mathematica in Action. New York: W. H. Freeman, pp. 353-362, 1991.Watkins,M. R. "Inexplicable Secrets of Creation." http://www.maths.ex.ac.uk/~mwatkins/zeta/.Weisstein, E. W. "Books about Riemann Zeta Function." http://www.ericweisstein.com/encyclopedias/books/RiemannZetaFunction.html.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Woon, S. C. "Generalization of a Relation Between the Riemann Zeta Function and Bernoulli Numbers." 24 Dec 1998. http://arxiv.org/abs/math.NT/9812143.Zucker, I. J. "The Summation of Series of Hyperbolic Functions." SIAM J. Math. Anal. 10, 192-206, 1979.Zucker, I. J. "Some Infinite Series of Exponential and Hyperbolic Functions." SIAM J. Math. Anal. 15, 406-413, 1984.Zudilin, W. "One of the Numbers zeta(5), zeta(7), zeta(9), zeta(11) Is Irrational." Uspekhi Mat. Nauk 56, 149-150, 2001.Zvengrowski, P. and Saidak, F. "On the Modulus of the Riemann Zeta Function in the Critical Strip." Math. Slovaca 53, 145-172, 2003.

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Sondow, Jonathan and Weisstein, Eric W. "Riemann Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannZetaFunction.html

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