Hurwitz zeta function

The Hurwitz zeta function has an integral representation

:$\backslash zeta(s,a)\; =\; \backslash frac\{1\}\{\backslash Gamma(s)\}\; \backslash int\_0^\backslash infty\; \backslash frac\{x^\{s-1\}e^\{-ax\}\}\{1-e^\{-x\}\}\; dx$

for $\backslash operatorname\{Re\}(s)>1$ and $\backslash operatorname\{Re\}(a)>0.$ (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing

:$\backslash zeta(s,a)\backslash Gamma(s)$

= \sum_{n=0}^\infty \frac{1}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x}

= \sum_{n=0}^\infty \int_0^\infty y^s e^{-(n+a)y} \frac{dy}{y}

and then interchanging the sum and integral.{{harvnb|Apostol|1976|p=251|loc=Theorem 12.2}}

The integral representation above can be converted to a contour integral representation

:$\backslash zeta(s,a)\; =\; -\backslash Gamma(1-s)\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_C\; \backslash frac\{(-z)^\{s-1\}e^\{-az\}\}\{1-e^\{-z\}\}\; dz$

where $C$ is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation $(-z)^\{s-1\}$. Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.{{harvnb|Whittaker|Watson|1927|p=266|loc=Section 13.13}}

The contour integral representation provides an analytic continuation of $\backslash zeta(s,a)$ to all $s\; \backslash ne\; 1$. At $s\; =\; 1$, it has a simple pole with residue $1$.{{harvnb|Apostol|1976|p=255|loc=Theorem 12.4}}

The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:{{harvnb|Apostol|1976|p=257|loc=Theorem 12.6}}

:$\backslash zeta(1-s,a)\; =\; \backslash frac\{\backslash Gamma(s)\}\{(2\backslash pi)^s\}\; \backslash left(\; e^\{-\backslash pi\; i\; s/2\}\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{e^\{2\backslash pi\; ina\}\}\{n^s\}\; +\; e^\{\backslash pi\; i\; s/2\}\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{e^\{-2\backslash pi\; ina\}\}\{n^s\}\; \backslash right),$

valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:{{harvnb|Apostol|1976|p=259|loc=Theorem 12.7}}

:$\backslash zeta(1-s)\; =\; \backslash frac\{2\backslash Gamma(s)\}\{(2\backslash pi)^\{s\}\}\; \backslash cos\backslash left(\backslash frac\{\backslash pi\; s\}\{2\}\backslash right)\; \backslash zeta(s)$

Hurwitz's formula can also be expressed as{{harvnb|Whittaker|Watson|1927|pp=268–269|loc=Section 13.15}}

:$\backslash zeta(s,a)\; =\; \backslash frac\{2\backslash Gamma(1-s)\}\{(2\backslash pi)^\{1-s\}\}\; \backslash left(\; \backslash sin\backslash left(\backslash frac\{\backslash pi\; s\}\{2\}\backslash right)\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash cos(2\backslash pi\; na)\}\{n^\{1-s\}\}\; +\; \backslash cos\backslash left(\backslash frac\{\backslash pi\; s\}\{2\}\backslash right)\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash sin(2\backslash pi\; na)\}\{n^\{1-s\}\}\; \backslash right)$

(for Re(s) < 0 and 0 < a ≤ 1).

Hurwitz's formula has a variety of different proofs.See the references in Section 4 of: {{cite journal |first1= S. |last1= Kanemitsu |first2= Y. |last2= Tanigawa |first3= H. |last3= Tsukada |first4= M. |last4= Yoshimoto |title= Contributions to the theory of the Hurwitz zeta-function |journal= Hardy-Ramanujan Journal |volume= 30| date= 2007 |pages= 31–55 |doi= 10.46298/hrj.2007.159 |zbl= 1157.11036|doi-access= free }} One proof uses the contour integration representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation.{{cite journal |first=N. J. |last=Fine |author-link= Nathan Fine |title= Note on the Hurwitz Zeta-Function |journal= Proceedings of the American Mathematical Society |volume= 2 |number= 3 |date= June 1951 |pages= 361–364 |doi= 10.2307/2031757 |jstor=2031757 |doi-access=free |zbl= 0043.07802}} These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral

:$\backslash zeta(s,a)\; =\; s\; \backslash int\_\{-a\}^\backslash infty\; \backslash frac\{\backslash lfloor\; x\; \backslash rfloor\; -\; x\; +\; \backslash frac\{1\}\{2\}\}\{(x+a)^\{s+1\}\}\; dx$

(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.{{cite journal |first= Bruce C. |last= Berndt |author-link= Bruce C. Berndt |title= On the Hurwitz zeta-function |journal= Rocky Mountain Journal of Mathematics |volume= 2 |number= 1 |date= Winter 1972 |pages= 151–158 |doi= 10.1216/RMJ-1972-2-1-151 |zbl= 0229.10023|doi-access= free }}

When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers $1\backslash leq\; m\; \backslash leq\; n$,

:$\backslash zeta\; \backslash left(1-s,\backslash frac\{m\}\{n\}\; \backslash right)\; =$

\frac{2\Gamma(s)}{ (2\pi n)^s }

\sum_{k=1}^n \left[\cos

\left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\;

\zeta \left( s,\frac {k}{n} \right)\right]

holds for all values of s.{{harvnb|Apostol|1976|p=261|loc=Theorem 12.8}}

This functional equation can be written as another equivalent form:

$\backslash zeta\; \backslash left(1-s,\backslash frac\{m\}\{n\}\; \backslash right)\; =\; \backslash frac\{\backslash Gamma(s)\}\{\; (2\backslash pi\; n)^s\}\; \backslash sum\_\{k=1\}^n\; \backslash left[e^\{\backslash frac\{\backslash pi\; is\}\{2\}\}e^\{-\backslash frac\{2\backslash pi\; ikm\}\{n\}\}\backslash zeta\; \backslash left(\; s,\backslash frac\; \{k\}\{n\}\; \backslash right)\; +\; e^\{-\backslash frac\{\backslash pi\; is\}\{2\}\}e^\{\backslash frac\{2\backslash pi\; ikm\}\{n\}\}\backslash zeta\; \backslash left(\; s,\backslash frac\; \{k\}\{n\}\; \backslash right)\; \backslash right]$

.

Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form

:$$

\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right)

\cos\dfrac{2\pi rk}{m} =\frac{m \Gamma(1-s)}{(2\pi m)^{1-s}}

\sin\frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) +

\zeta\left(1-s,1-\frac{k}{m}\right) \right\} - \zeta(s)

:$$

\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right)

\sin\dfrac{2\pi rk}{m}= \frac{m \Gamma(1-s)}{(2\pi m)^{1-s}}

\cos \frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) -

\zeta\left(1-s,1-\frac{k}{m}\right)\right\}

:$$

\sum_{r=1}^{m-1} \zeta^2\left(s,\frac{r}{m}\right) =

\big(m^{2s-1}-1 \big)\zeta^2(s) + \frac{2m\Gamma^2(1-s)}{(2\pi m)^{2-2s}}

\sum_{l=1}^{m-1} \left\{\zeta\left(1-s,\frac{l}{m}\right) - \cos\pi s

\cdot \zeta\left(1-s,1-\frac{l}{m}\right)\right\} \zeta\left(1-s,\frac{l}{m}\right)

where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.{{cite journal|doi=10.1016/j.jnt.2014.08.009 |first=I.V. |last=Blagouchine |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |publisher=Elsevier |volume=148 |pages=537–592 |date=2014 |arxiv=1401.3724}}

A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:{{Citation |first=Helmut |last=Hasse |title=Ein Summierungsverfahren für die Riemannsche ζ-Reihe |year=1930 |journal=Mathematische Zeitschrift |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 | jfm=56.0894.03 |s2cid=120392534 | url=https://eudml.org/doc/168238 }}

:$\backslash zeta(s,a)=\backslash frac\{1\}\{s-1\}$

\sum_{n=0}^\infty \frac{1}{n+1}

\sum_{k=0}^n (-1)^k {n \choose k} (a+k)^{1-s}.

This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of $a^\{1-s\}$; that is,

:$\backslash Delta^n\; a^\{1-s\}\; =\; \backslash sum\_\{k=0\}^n\; (-1)^\{n-k\}\; \{n\; \backslash choose\; k\}\; (a+k)^\{1-s\}$

where Δ is the forward difference operator. Thus, one may write:

:$\backslash begin\{align\}$

\zeta(s, a) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n a^{1-s}\\

&= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} a^{1-s}

\end{align}

The partial derivative of the zeta in the second argument is a shift:

:$\backslash frac\; \{\backslash partial\}\; \{\backslash partial\; a\}\; \backslash zeta\; (s,a)\; =\; -s\backslash zeta(s+1,a).$

Thus, the Taylor series can be written as:

:$\backslash zeta(s,x+y)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\; \{y^k\}\; \{k!\}$

\frac {\partial^k} {\partial x^k} \zeta (s,x) =

\sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).

Alternatively,

:$\backslash zeta(s,\; q)\; =\; \backslash frac\{1\}\{q^s\}\; +\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; (-q)^n\; \{s\; +\; n\; -\; 1\; \backslash choose\; n\}\; \backslash zeta(s\; +\; n),$

with .{{cite journal |last=Vepstas |first=Linas |arxiv=math/0702243 |title=An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions |year=2007 |doi=10.1007/s11075-007-9153-8 |volume=47 |issue=3 |journal=Numerical Algorithms |pages=211–252|bibcode=2008NuAlg..47..211V|s2cid=15131811 }}

Closely related is the Stark–Keiper formula:

:$\backslash zeta(s,N)\; =$

\sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right]

{s+k-1 \choose s-1} (-1)^k \zeta (s+k,N)

which holds for integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers.

The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series

:$\backslash zeta(s,a)\; =\; \backslash frac\{1\}\{s-1\}\; +\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n!\}\; \backslash gamma\_n(a)\; (s-1)^n.$

In particular, the constant term is given by

:$\backslash lim\_\{s\backslash to\; 1\}\; \backslash left[\; \backslash zeta(s,a)\; -\; \backslash frac\{1\}\{s-1\}\backslash right]\; =$

\gamma_0(a)=

\frac{-\Gamma'(a)}{\Gamma(a)} = -\psi(a)

where $\backslash Gamma$ is the gamma function and $\backslash psi\; =\; \backslash Gamma\text{'}\; /\; \backslash Gamma$ is the digamma function. As a special case, $\backslash gamma\_0(1)\; =\; -\backslash psi(1)\; =\; \backslash gamma\_0\; =\; \backslash gamma$.

The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function.{{cite journal |last=Jacek Klinowski |first=Djurdje Cvijović | title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments |journal=Mathematics of Computation |volume=68 |date=1999|issue=228 |pages=1623–1631 |doi=10.1090/S0025-5718-99-01091-1|bibcode=1999MaCom..68.1623C |doi-access=free }}

The values of ζ(s, a) at s = 0, −1, −2, ... are related to the Bernoulli polynomials:{{harvnb|Apostol|1976|p=264|loc=Theorem 12.13}}

:$\backslash zeta(-n,a)\; =\; -\backslash frac\{B\_\{n+1\}(a)\}\{n+1\}.$

For example, the $n=0$ case gives{{harvnb|Apostol|1976|p=268}}

:$\backslash zeta(0,a)\; =\; \backslash frac\{1\}\{2\}\; -\; a.$

The partial derivative with respect to s at s = 0 is related to the gamma function:

:$\backslash left.\; \backslash frac\{\backslash partial\}\{\backslash partial\; s\}\; \backslash zeta(s,a)\; \backslash right|\_\{s=0\}\; =\; \backslash log\backslash Gamma(a)\; -\; \backslash frac\{1\}\{2\}\; \backslash log(2\backslash pi)$

In particular, $\backslash zeta\text{'}(0)\; =\; -\backslash frac\{1\}\{2\}\; \backslash log(2\backslash pi).$ The formula is due to Lerch.{{cite journal |last=Berndt |first=Bruce C. |author-link=Bruce C. Berndt |title=The Gamma Function and the Hurwitz Zeta-Function |journal=The American Mathematical Monthly |volume=92 |number=2 |date=1985 |pages=126–130 |doi=10.2307/2322640|jstor=2322640 }}{{harvnb|Whittaker|Watson|1927|p=271|loc=Section 13.21}}

If $\backslash vartheta\; (z,\backslash tau)$ is the Jacobi theta function, then

:$\backslash int\_0^\backslash infty\; \backslash left[\backslash vartheta\; (z,it)\; -1\; \backslash right]\; t^\{s/2\}\; \backslash frac\{dt\}\{t\}=$

\pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right)

\left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]

holds for $\backslash Re\; s\; >\; 0$ and z complex, but not an integer. For z=n an integer, this simplifies to

:$\backslash int\_0^\backslash infty\; \backslash left[\backslash vartheta\; (n,it)\; -1\; \backslash right]\; t^\{s/2\}\; \backslash frac\{dt\}\{t\}=$

2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s)

=2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).

where ζ here is the Riemann zeta function. Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function, or Dirac comb in z as $t\backslash rightarrow\; 0$.

At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when a = 1, when a = 1/2 it is equal to (2^s−1)ζ(s), and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then{{cite web|last=Lowry|first=David|title=Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa|url=http://mixedmath.wordpress.com/2013/02/08/hurwitz-zeta-is-a-sum-of-dirichlet-l-functions-and-vice-versa/|work=mixedmath|date=8 February 2013|access-date=8 February 2013}}

:$\backslash zeta(s,n/k)=\backslash frac\{k^s\}\{\backslash varphi(k)\}\backslash sum\_\backslash chi\backslash overline\{\backslash chi\}(n)L(s,\backslash chi),$

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination

:$L(s,\backslash chi)=\backslash frac\; \{1\}\{k^s\}\; \backslash sum\_\{n=1\}^k\; \backslash chi(n)\backslash ;\; \backslash zeta\; \backslash left(s,\backslash frac\{n\}\{k\}\backslash right).$

There is also the multiplication theorem

:$k^s\backslash zeta(s)=\backslash sum\_\{n=1\}^k\; \backslash zeta\backslash left(s,\backslash frac\{n\}\{k\}\backslash right),$

of which a useful generalization is the distribution relation{{cite book | first1=Daniel S. | last1=Kubert | author-link1=Daniel Kubert | first2=Serge | last2=Lang | author-link2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | publisher=Springer-Verlag | year=1981 | isbn=0-387-90517-0 | zbl=0492.12002 | page=13 }}

:$\backslash sum\_\{p=0\}^\{q-1\}\backslash zeta(s,a+p/q)=q^s\backslash ,\backslash zeta(s,qa).$

(This last form is valid whenever q a natural number and 1 − qa is not.)

If a=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if a=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<a<1 and a≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn for rational or transcendental irrational a,{{Citation |last1=Davenport |first1=H. |name-list-style=amp |last2=Heilbronn |first2=H. |title=On the zeros of certain Dirichlet series |journal=Journal of the London Mathematical Society |volume=11 |issue=3 |year=1936 |pages=181–185 |doi=10.1112/jlms/s1-11.3.181 | zbl=0014.21601 }} and by Cassels for algebraic irrational a.Davenport (1967) p.73{{Citation |last=Cassels |first=J. W. S. |title=Footnote to a note of Davenport and Heilbronn |journal=Journal of the London Mathematical Society |volume=36 |issue=1 |year=1961 |pages=177–184 |doi=10.1112/jlms/s1-36.1.177 | zbl=0097.03403 }}

The Hurwitz zeta function occurs in a number of striking identities at rational values.Given by {{Citation |first1=Djurdje |last1=Cvijović |name-list-style=amp |first2=Jacek |last2=Klinowski |title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments |journal=Mathematics of Computation |volume=68 |issue=228 |year=1999 |pages=1623–1630 |doi=10.1090/S0025-5718-99-01091-1|bibcode=1999MaCom..68.1623C |doi-access=free }} In particular, values in terms of the Euler polynomials $E\_n(x)$:

:$E\_\{2n-1\}\backslash left(\backslash frac\{p\}\{q\}\backslash right)\; =$

(-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}}

\sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right)

\cos \frac{(2k-1)\pi p}{q}

and

:$E\_\{2n\}\backslash left(\backslash frac\{p\}\{q\}\backslash right)\; =$

(-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}}

\sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right)

\sin \frac{(2k-1)\pi p}{q}

One also has

:$\backslash zeta\backslash left(s,\backslash frac\{2p-1\}\{2q\}\backslash right)\; =$

2(2q)^{s-1} \sum_{k=1}^q \left[

C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) +

S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right)

\right]

which holds for $1\backslash le\; p\; \backslash le\; q$. Here, the $C\_\backslash nu(x)$ and $S\_\backslash nu(x)$ are defined by means of the Legendre chi function $\backslash chi\_\backslash nu$ as

:$C\_\backslash nu(x)\; =\; \backslash operatorname\{Re\}\backslash ,\; \backslash chi\_\backslash nu\; (e^\{ix\})$

and

:$S\_\backslash nu(x)\; =\; \backslash operatorname\{Im\}\backslash ,\; \backslash chi\_\backslash nu\; (e^\{ix\}).$

For integer values of ν, these may be expressed in terms of the Euler polynomials. These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics, it occurs in Zipf's law and the Zipf–Mandelbrot law. In particle physics, it occurs in a formula by Julian Schwinger,{{Citation |last=Schwinger |first=J. |title=On gauge invariance and vacuum polarization |journal=Physical Review |volume=82 |issue=5 |year=1951 |pages=664–679 |doi=10.1103/PhysRev.82.664 |bibcode=1951PhRv...82..664S}} giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

The Hurwitz zeta function with a positive integer m is related to the polygamma function:

:$\backslash psi^\{(m)\}(z)=\; (-1)^\{m+1\}\; m!\; \backslash zeta\; (m+1,z)\; \backslash \; .$

The Barnes zeta function generalizes the Hurwitz zeta function.

The Lerch transcendent generalizes the Hurwitz zeta:

:$\backslash Phi(z,\; s,\; q)\; =\; \backslash sum\_\{k=0\}^\backslash infty$

\frac { z^k} {(k+q)^s}

and thus

:$\backslash zeta(s,a)=\backslash Phi(1,\; s,\; a).\backslash ,$

Hypergeometric function

:$\backslash zeta(s,a)=a^\{-s\}\backslash cdot\{\}\_\{s+1\}F\_s(1,a\_1,a\_2,\backslash ldots\; a\_s;a\_1+1,a\_2+1,\backslash ldots\; a\_s+1;1)$ where $a\_1=a\_2=\backslash ldots=a\_s=a\backslash text\{\; and\; \}a\backslash notin\backslash N\backslash text\{\; and\; \}s\backslash in\backslash N^+.$

Meijer G-function

:$\backslash zeta(s,a)=G\backslash ,\_\{s+1,\backslash ,s+1\}^\{\backslash ,1,\backslash ,s+1\}\backslash left(-1\; \backslash ;\; \backslash left|\; \backslash ;\; \backslash begin\{matrix\}0,1-a,\backslash ldots,1-a\backslash \backslash 0,-a,\backslash ldots,-a\backslash end\{matrix\}\backslash right)\backslash right.\backslash qquad\backslash qquad\; s\backslash in\backslash N^+.$

• {{dlmf|id=25.11|first=T. M. |last=Apostol}}
• See chapter 12 of {{Apostol IANT}}
• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. {{ISBN|0-486-61272-4}}. (See Paragraph 6.4.10 for relationship to polygamma function.)
• {{cite book | last=Davenport | first=Harold | author-link=Harold Davenport | title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
• {{cite journal

|first1=Jeff

|last1=Miller

|first2=Victor S.

|last2=Adamchik

|title= Derivatives of the Hurwitz Zeta Function for Rational Arguments

|journal= Journal of Computational and Applied Mathematics

|volume=100

|issue=2

|year=1998

|pages=201–206

|doi=10.1016/S0377-0427(98)00193-9

|doi-access=free

}}

• {{cite book

|title=A Course Of Modern Analysis

|title-link=A Course of Modern Analysis

|author-last1=Whittaker

|author-first1=E. T.

|author-link1=Edmund Taylor Whittaker

|author-last2=Watson

|author-first2=G. N.

|author-link2=George Neville Watson

|date=1927

|edition=4th

|publisher=Cambridge University Press

|publication-place=Cambridge, UK

}}

• {{mathworld|urlname=HurwitzZetaFunction|title=Hurwitz Zeta Function|author=Jonathan Sondow and Eric W. Weisstein}}

Category:Zeta and L-functions