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Eta Function Analytic Continuation Zeta Function

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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

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

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

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

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

(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

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

(OEIS A114875).

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

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

(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

(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

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,

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

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

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

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

(Plouffe 1998).

Another set of related formulas are

(Plouffe 2006).

Multiterm sums for odd zeta(n) include

(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

Sums involving integers multiples of the argument include

where H_n is a harmonic number.

Two surprising sums involving zeta(x) are given by

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

(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.

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

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