Hurwitz Zeta Function

A generalization of the Riemann Zeta Function with a Formula

$$\zeta(s,a) \equiv \sum_{k=0}^\infty {1\over (k+a)^s},$$ (1)

where any term with $k+a=0$ is excluded. The Hurwitz zeta function can also be given by the functional equation

$$\zeta\left({s,{p\over q}}\right)=2\Gamma(1-s)(2\pi q)^{s-1}\... ...+{2\pi np\over q}}\right)\zeta\left({\!1-s, {n\over q}}\right)$$ (2)

(Apostol 1976, Miller and Adamchik), or the integral

$$\zeta(s,a)= {\textstyle{1\over 2}}a^{-3}+{a^{1-s}\over s-1}+... ...eft({y\over a}\right)}\right]}\right\} {dy\over e^{2\pi y}-1}.$$ (3)

If $\Re[z]<0$, then

$$\zeta(z,a) = {2\Gamma(1-z)\over (2\pi)^{1-z}}\left[{\sin\lef... ...}\right)\sum_{n=1}^\infty{\sin(2\pi an)\over n^{1-z}}}\right].$$ (4)

The Hurwitz zeta function satisfies

$$\zeta(0,a) = {\textstyle{1\over 2}}-a$$ (5)

$${d\over ds} \zeta(0,a) = \ln[\Gamma(a)]-{\textstyle{1\over 2}}\ln(2\pi)$$ (6)

$${d\over ds} \zeta(0,0) = {\textstyle{1\over 2}}\ln(2\pi),$$ (7)

where $\Gamma(z)$ is the Gamma Function. The Polygamma Function $\psi_m(z)$ can be expressed in terms of the Hurwitz zeta function by

$$\psi_m(z)=(-1)^{m+1}m!\zeta(1+m,z).$$ (8)

For Positive integers $k$, $p$, and $q>p$,

$\zeta'\left({-2k+1, {p\over q}}\right)= {[\psi(2k)-\ln(2\pi q)]B_{2k}(p/q)\over 2k}-{[\psi(2k)-\ln(2\pi)]B_{2k}\over q^{2k}2k}$
$\quad +{(-1)^{k+1}\pi\over(2\pi q)^{2k}}\sum_{n=1}^{q-1} \sin\left({2\pi pn\over q}\right)\psi_{(2k-1)}\left({n\over q}\right)$
$\quad +{(-1)^{k+1}2(2k-1)!\over(2\pi q)^{2k}}\sum_{n=1}^{q-1}\cos\left({2\pi pn\over q}\right)\zeta'\left({2k, {n\over q}}\right)+{\zeta'(-2k+1)\over q^{2k}},$	(9)

where $B_n$ is a Bernoulli Number, $B_n(x)$ a Bernoulli Polynomial, $\psi_n(z)$ is a Polygamma Function, and $\zeta(z)$ is a Riemann Zeta Function (Miller and Adamchik). Miller and Adamchik also give the closed-form expressions

$\zeta'(-2k+1, {\textstyle{1\over 2}})=-{B_{2k}\ln 2\over 4^k k}-{(2^{2k-1}-1)\zeta'(-2k+1)\over 2^{2k-1}}$	(10)
$\zeta'\left({-2k+1, {{\textstyle{1\over 3}}\atop{\textstyle{2\over 3}}}}\right)... ... 3}})\over 2\sqrt{3}(6\pi)^{2k-1}}-{(3^{2k-1}-1)\zeta'(-2k+1)\over 2(3^{2k-1})}$	(11)
$\zeta'\left({\!-2k+1,{{\textstyle{1\over 4}}\atop{\textstyle{3\over 4}}}}\right... ...tyle{1\over 4}})\over 4(8\pi)^{2k-1}}-{(2^{2k-1}-1)\zeta'(-2k+1)\over 2^{4k-1}}$	(12)
$\zeta'\left({-2k+1, {{\textstyle{1\over 6}}\atop{\textstyle{5\over 6}}}}\right)... ...}(3^{2k-1}-1)\ln 2\over(6^{2k-1})4k}+{B_{2k}(2^{2k-1}-1)\ln 3\over(6^{2k-1})4k}$
$\quad \mp{(-1)^k(2^{2k-1}+1)\psi_{2k-1}({\textstyle{1\over 3}})\over 2\sqrt{3}\,(12\pi)^{2k-1}}+{(2^{2k-1}-1)(3^{2k-1}-1)\zeta'(-2k+1)\over 2(6^{2k-1})}.$	(13)

See also Khintchine's Constant, Polygamma Function, Psi Function, Riemann Zeta Function, Zeta Function

References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995.

Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regularization Techniques with Applications. River Edge, NJ: World Scientific, 1994.

Knopfmacher, J. ``Generalised Euler Constants.'' Proc. Edinburgh Math. Soc. 21, 25-32, 1978.

Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966.

Miller, J. and Adamchik, V. ``Derivatives of the Hurwitz Zeta Function for Rational Arguments.'' Submitted to J. Symb. Comput.

Spanier, J. and Oldham, K. B. ``The Hurwitz Function $\zeta(\nu;u)$.'' Ch. 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 653-664, 1987.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 268-269, 1950.