Lerch zeta function

In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.

Definition
The Lerch zeta function is given by


 * $$L(\lambda, s, \alpha) = \sum_{n=0}^\infty

\frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}.$$

A related function, the Lerch transcendent, is given by


 * $$\Phi(z, s, \alpha) = \sum_{n=0}^\infty

\frac { z^n} {(n+\alpha)^s}$$.

The transcendent only converges for any real number $$\alpha > 0$$, where:

$$|z| < 1$$, or

$$\mathfrak{R}(s) > 1$$, and $$|z| = 1$$.

The two are related, as


 * $$\,\Phi(e^{2\pi i\lambda}, s,\alpha)=L(\lambda, s, \alpha).$$

Integral representations
The Lerch transcendent has an integral representation:

\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt$$ The proof is based on using the integral definition of the Gamma function to write
 * $$\Phi(z,s,a)\Gamma(s)

= \sum_{n=0}^\infty \frac{z^n}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x} = \sum_{n=0}^\infty \int_0^\infty t^s z^n e^{-(n+a)t} \frac{dt}{t}$$ and then interchanging the sum and integral. The resulting integral representation converges for $$z \in \Complex \setminus [1,\infty),$$ Re(s) > 0, and Re(a) > 0. This analytically continues $$\Phi(z,s,a)$$ to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.

A contour integral representation is given by

\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt$$ where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points $$t = \log(z) + 2k\pi i$$ (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.

Other integral representations
A Hermite-like integral representation is given by



\Phi(z,s,a)= \frac{1}{2a^s}+ \int_0^\infty \frac{z^t}{(a+t)^s}\,dt+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt $$ for
 * $$\Re(a)>0\wedge |z|<1 $$

and

\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{\log^{s-1}(1/z)}{z^a}\Gamma(1-s,a\log(1/z))+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt $$ for
 * $$\Re(a)>0. $$

Similar representations include



\Phi(z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \tanh\pi t }\,dt, $$

and


 * $$\Phi(-z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \sinh\pi t }\,dt,$$

holding for positive z (and more generally wherever the integrals converge). Furthermore,


 * $$\Phi(e^{i\varphi},s,a)=L\big(\tfrac{\varphi}{2\pi}, s, a\big)= \frac{1}{a^s} + \frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}\big(e^{i\varphi}-e^{-t}\big)}{\cosh{t}-\cos{\varphi}}\,dt,$$

The last formula is also known as Lipschitz formula.

Special cases
The Lerch zeta function and Lerch transcendent generalize various special functions.

The Hurwitz zeta function is the special case
 * $$\zeta(s,\alpha) = L(0, s, \alpha) = \Phi(1,s,\alpha) = \sum_{n=0}^\infty \frac{1}{(n+\alpha)^s}.$$

The polylogarithm is another special case:
 * $$\textrm{Li}_s(z)=z\Phi(z,s,1) = \sum_{n=1}^\infty \frac{z^n}{n^s}.$$

The Riemann zeta function is a special case of both of the above:
 * $$\zeta(s) = \Phi(1,s,1) = \sum_{n=1}^\infty \frac{1}{n^s}$$

Other special cases include:
 * The Dirichlet eta function:
 * $$\eta(s) = \Phi(-1,s,1) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$


 * The Dirichlet beta function:
 * $$\beta(s) = 2^{-s} \Phi(-1,s,1/2) = \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)^s}$$


 * The Legendre chi function:
 * $$\chi_s(z)=2^{-s}z \Phi(z^2,s,1/2) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^s}$$


 * The polygamma function:
 * $$\psi^{(n)}(\alpha)= (-1)^{n+1} n!\Phi (1,n+1,\alpha)$$

Identities
For λ rational, the summand is a root of unity, and thus $$L(\lambda, s, \alpha)$$ may be expressed as a finite sum over the Hurwitz zeta function. Suppose $\lambda = \frac{p}{q}$ with $$p, q \in \Z$$ and $$q > 0$$. Then $$z = \omega = e^{2 \pi i \frac{p}{q}}$$ and $$\omega^q = 1$$.


 * $$\Phi(\omega, s, \alpha) = \sum_{n=0}^\infty

\frac {\omega^n} {(n+\alpha)^s} = \sum_{m=0}^{q-1} \sum_{n=0}^\infty \frac {\omega^{qn + m}}{(qn + m + \alpha)^s} = \sum_{m=0}^{q-1} \omega^m q^{-s} \zeta \left( s,\frac{m + \alpha}{q} \right) $$

Various identities include:
 * $$\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}$$

and


 * $$\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)$$

and


 * $$\Phi(z,s+1,a)=-\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).$$

Series representations
A series representation for the Lerch transcendent is given by


 * $$\Phi(z,s,q)=\frac{1}{1-z}

\sum_{n=0}^\infty \left(\frac{-z}{1-z} \right)^n \sum_{k=0}^n (-1)^k \binom{n}{k} (q+k)^{-s}.$$ (Note that $$\tbinom{n}{k}$$ is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)&lt;1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for
 * $$\left|\log(z)\right| < 2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots$$

\Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1} +\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right] $$

If n is a positive integer, then

\Phi(z,n,a)=z^{-a}\left\{ \sum_{{k=0}\atop k\neq n-1}^ \infty \zeta(n-k,a)\frac{\log^k (z)}{k!} +\left[\psi(n)-\psi(a)-\log(-\log(z))\right]\frac{\log^{n-1}(z)}{(n-1)!} \right\}, $$ where $$\psi(n)$$ is the digamma function.

A Taylor series in the third variable is given by
 * $$\Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};|x|<\Re(a),$$

where $$(s)_{k}$$ is the Pochhammer symbol.

Series at a = −n is given by

\Phi(z,s,a)=\sum_{k=0}^n \frac{z^k}{(a+k)^s} +z^n\sum_{m=0}^\infty (1-m-s)_{m}\operatorname{Li}_{s+m}(z)\frac{(a+n)^m}{m!};\ a\rightarrow-n $$

A special case for n = 0 has the following series

\Phi(z,s,a)=\frac{1}{a^s} +\sum_{m=0}^\infty (1-m-s)_m \operatorname{Li}_{s+m}(z)\frac{a^m}{m!}; |a|<1, $$ where $$\operatorname{Li}_s(z)$$ is the polylogarithm.

An asymptotic series for $$s\rightarrow-\infty$$
 * $$\Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [2k\pi i-\log(z)]^{s-1}e^{2k\pi ai}

$$ for $$|a|<1;\Re(s)<0 ;z\notin (-\infty,0) $$ and

\Phi(-z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai} $$ for $$|a|<1;\Re(s)<0 ;z\notin (0,\infty). $$

An asymptotic series in the incomplete gamma function

\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{1}{z^a}\sum_{k=1}^\infty \frac{e^{-2\pi i(k-1)a}\Gamma(1-s,a(-2\pi i(k-1)-\log(z)))} {(-2\pi i(k-1)-\log(z))^{1-s}}+ \frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}} $$ for $$|a|<1;\Re(s)<0.$$

The representation as a generalized hypergeometric function is

\Phi(z,s,\alpha)=\frac{1}{\alpha^s}{}_{s+1}F_s\left(\begin{array}{c} 1,\alpha,\alpha,\alpha,\cdots\\ 1+\alpha,1+\alpha,1+\alpha,\cdots\\ \end{array}\mid z\right). $$

Asymptotic expansion
The polylogarithm function $$\mathrm{Li}_n(z)$$ is defined as
 * $$\mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z).$$

Let

\Omega_{a} \equiv\begin{cases} \mathbb{C}\setminus[1,\infty) & \text{if } \Re a > 0, \\ {z \in \mathbb{C}, |z|<1} & \text{if } \Re a \le 0. \end{cases} $$ For $$|\mathrm{Arg}(a)|<\pi, s \in \mathbb{C}$$ and $$z \in \Omega_{a}$$, an asymptotic expansion of $$\Phi(z,s,a)$$ for large $$a$$ and fixed $$s$$ and $$z$$ is given by

\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} +   \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s}) $$ for $$N \in \mathbb{N}$$, where $$(s)_n = s (s+1)\cdots (s+n-1)$$ is the Pochhammer symbol.

Let
 * $$f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}.$$

Let $$C_{n}(z,a)$$ be its Taylor coefficients at $$x=0$$. Then for fixed $$N \in \mathbb{N}, \Re a > 1$$ and $$\Re s > 0$$,

\Phi(z,s,a) - \frac{\mathrm{Li}_{s}(z)}{z^{a}} = \sum_{n=0}^{N-1} C_{n}(z,a) \frac{(s)_{n}}{a^{n+s}} + O\left( (\Re a)^{1-N-s}+a z^{-\Re a} \right), $$ as $$\Re a \to \infty$$.

Software
The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.