Hurwitz Zeta Function

The Hurwitz zeta function is a generalization of the Riemann zeta function that is also known as the generalized zeta function. It is classically defined by the formula

(1)

for and by analytic continuation to other , where any term with is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].

The slightly different form

(2)

is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for .

The plot above shows for real and , with the zero contour indicated in black.

For , a globally convergent series for (which, for fixed , gives an analytic continuation of to the entire complex -plane except the point ) is given by

(3)

(Hasse 1930).

The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].

For , reduces to the Riemann zeta function ,

(4)

If the singular term is excluded from the sum definition of , then as well.

The Hurwitz zeta function is given by the integral

(5)

for and .

The plot above illustrates the complex zeros of (Trott 1999), where . Here, the complex -plane is horizontal and the real -line is vertical and runs from at the bottom to at the top. The upper line is the critical line , which contains zeros of . The lower two lines are and (again), which contain zeros of and , respectively, since ; cf. equation (9) below.

This plot also appeared on the cover of the March 2004 issue of FOCUS, the Mathematical Association of America's news magazine.

The Hurwitz zeta function can also be given by the functional equation

(6)

(Apostol 1995, Miller and Adamchik 1999), or the integral

$zeta(s,a)=1/2a^(-s)+(a^(1-s))/(s-1)+2int_0^infty(a^2+y^2)^(-s/2){sin[stan^(-1)(y/a)]}(dy)/(e^(2piy)-1).$

(7)

If and , then

(8)

(Hurwitz 1882; Whittaker and Watson 1990, pp. 268-269).

The Hurwitz zeta function satisfies

(9)

for (Apostol 1995, p. 264), where is a Bernoulli polynomial, giving the special case

(10)

In addition,

			(11)
			(12)
			(13)
			(14)
			(15)

Derivative identities include

			(16)
			(17)

where is the gamma function (Bailey et al. 2006, p. 179). The definition (1) implies that

(18)

for .

In the limit,

(19)

(Whittaker and Watson 1990, p. 271; Allouche 1992), where is the digamma function.

The polygamma function can be expressed in terms of the Hurwitz zeta function by

(20)

For positive integers , , and ,

zeta^'(-2k+1,p/q)=([psi(2k)-ln(2piq)]B_(2k)(p/q))/(2k)-([psi(2k)-ln(2pi)]B_(2k))/(q^(2k)2k)+((-1)^(k+1)pi)/((2piq)^(2k))sum_(n=1)^(q-1)sin((2pipn)/q)psi_((2k-1))(n/q)+((-1)^(k+1)2(2k-1)!)/((2piq)^(2k))sum_(n=1)^(q-1)cos((2pipn)/q)zeta^'(2k,n/q)+(zeta^'(-2k+1))/(q^(2k)),

(21)

where is a Bernoulli number, a Bernoulli polynomial, is a polygamma function, and is the Riemann zeta function (Miller and Adamchik 1999). Miller and Adamchik (1999) also give the closed-form expressions (where a large number of typos have been corrected in the expressions below)

			(22)
			(23)
			(24)
			(25)

where means , means , and the upper and lower fractions on the left side of the equations correspond to the plus and minus signs, respectively, on the right side.

Hurwitz Zeta Function

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