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

$L(\lambda ,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{e^{2\pi i\lambda n}}{(n+\alpha {)}^s}.$

A related function, the Lerch transcendent, is given by

$\mathrm{\Phi }(z,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{(n+\alpha {)}^s}$.

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

$|z|<1$, or

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

The two are related, as

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

Integral representations

The Lerch transcendent has an integral representation:

$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t^{s-1}{e}^{-at}}{1-z{e}^{-t}}dt$

The proof is based on using the integral definition of the Gamma function to write

$\mathrm{\Phi }(z,s,a)\mathrm{\Gamma }(s)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{(n+a{)}^s}\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^s{e}^{-x}\frac{dx}{x}=\sum _{n=0}^{\mathrm{\infty }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^s{z}^n{e}^{-(n+a)t}\frac{dt}{t}$

and then interchanging the sum and integral. The resulting integral representation converges for $z\in \mathbb{ℂ}\setminus [1,\mathrm{\infty }),$ Re(s) > 0, and Re(a) > 0. This analytically continues $\mathrm{\Phi }(z,s,a)$ 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

$\mathrm{\Phi }(z,s,a)=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}\underset{C}{\int }\frac{(-t{)}^{s-1}{e}^{-at}}{1-z{e}^{-t}}dt$

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

$\mathrm{\Phi }(z,s,a)=\frac{1}{2a^s}+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{z^t}{(a+t{)}^s}dt+\frac{2}{a^{s-1}}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (s\arctan (t)-ta\log (z))}{(1+t^2{)}^{s/2}(e^{2\pi at}-1)}dt$

for

$\mathrm{\Re }(a)>0\wedge |z|<1$

and

$\mathrm{\Phi }(z,s,a)=\frac{1}{2a^s}+\frac{{\log }^{s-1}(1/z)}{z^a}\mathrm{\Gamma }(1-s,a\log (1/z))+\frac{2}{a^{s-1}}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (s\arctan (t)-ta\log (z))}{(1+t^2{)}^{s/2}(e^{2\pi at}-1)}dt$

for

$\mathrm{\Re }(a)>0.$

Similar representations include

$\mathrm{\Phi }(z,s,a)=\frac{1}{2a^s}+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (t\log z)\sin (s\arctan \frac{t}{a})-\sin (t\log z)\cos (s\arctan \frac{t}{a})}{(a^2+t^2{)}^{\frac{s}{2}}\tanh \pi t}dt,$

and

$\mathrm{\Phi }(-z,s,a)=\frac{1}{2a^s}+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (t\log z)\sin (s\arctan \frac{t}{a})-\sin (t\log z)\cos (s\arctan \frac{t}{a})}{(a^2+t^2{)}^{\frac{s}{2}}\sinh \pi t}dt,$

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

$\mathrm{\Phi }(e^{i\phi },s,a)=L(\frac{\phi }{2\pi },s,a)=\frac{1}{a^s}+\frac{1}{2\mathrm{\Gamma }(s)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t^{s-1}{e}^{-at}(e^{i\phi }-e^{-t})}{\cosh t-\cos \phi }dt,$

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

$\zeta (s,\alpha )=L(0,s,\alpha )=\mathrm{\Phi }(1,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(n+\alpha {)}^s}.$

The polylogarithm is another special case:

${\text{<mrow data-mjx-texclass="ORD">Li</mrow>}}_s(z)=z\mathrm{\Phi }(z,s,1)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$

The Riemann zeta function is a special case of both of the above:

$\zeta (s)=\mathrm{\Phi }(1,s,1)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$

Other special cases include:

• The Dirichlet eta function:

$\eta (s)=\mathrm{\Phi }(-1,s,1)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{n^s}$

• The Dirichlet beta function:

$\beta (s)=2^{-s}\mathrm{\Phi }(-1,s,1/2)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{(2k+1{)}^s}$

• The Legendre chi function:

${\chi }_s(z)=2^{-s}z\mathrm{\Phi }(z^2,s,1/2)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^{2k+1}}{(2k+1{)}^s}$

• The polygamma function:

${\psi }^{(n)}(\alpha )=(-1{)}^{n+1}n!\mathrm{\Phi }(1,n+1,\alpha )$

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 $p,q\in \mathbb{\mathbb{Z}}$ and $q>0$. Then $z=\omega =e^{2\pi i\frac{p}{q}}$ and ${\omega }^q=1$.

$\mathrm{\Phi }(\omega ,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{{\omega }^n}{(n+\alpha {)}^s}=\sum _{m=0}^{q-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\omega }^{qn+m}}{(qn+m+\alpha {)}^s}=\sum _{m=0}^{q-1}{\omega }^m{q}^{-s}\zeta (s,\frac{m+\alpha }{q})$

Various identities include:

$\mathrm{\Phi }(z,s,a)=z^n\mathrm{\Phi }(z,s,a+n)+\sum _{k=0}^{n-1}\frac{z^k}{(k+a{)}^s}$

and

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

and

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

Series representations

A series representation for the Lerch transcendent is given by

$\mathrm{\Phi }(z,s,q)=\frac{1}{1-z}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{-z}{1-z}\right)}^n\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})(q+k{)}^{-s}.$

(Note that $(\genfrac{}{}{0ex}{}{n}{k})$ is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)<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

$|\log (z)|<2\pi ;s\ne 1,2,3,\dots ;a\ne 0,-1,-2,\dots$

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

If n is a positive integer, then

$\mathrm{\Phi }(z,n,a)=z^{-a}\{\sum _{\genfrac{}{}{0ex}{}{k=0}{k\ne n-1}}^{\mathrm{\infty }}\zeta (n-k,a)\frac{{\log }^k(z)}{k!}+[\psi (n)-\psi (a)-\log (-\log (z))]\frac{{\log }^{n-1}(z)}{(n-1)!}\},$

where $\psi (n)$ is the digamma function.

A Taylor series in the third variable is given by

$\mathrm{\Phi }(z,s,a+x)=\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Phi }(z,s+k,a)(s{)}_k\frac{(-x{)}^k}{k!};|x|<\mathrm{\Re }(a),$

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

Series at a = −n is given by

$\mathrm{\Phi }(z,s,a)=\sum _{k=0}^n\frac{z^k}{(a+k{)}^s}+z^n\sum _{m=0}^{\mathrm{\infty }}(1-m-s{)}_m{\mathrm{Li}}_{s+m}(z)\frac{(a+n{)}^m}{m!};a\to -n$

A special case for n = 0 has the following series

$\mathrm{\Phi }(z,s,a)=\frac{1}{a^s}+\sum _{m=0}^{\mathrm{\infty }}(1-m-s{)}_m{\mathrm{Li}}_{s+m}(z)\frac{a^m}{m!};|a|<1,$

where ${\mathrm{Li}}_s(z)$ is the polylogarithm.

An asymptotic series for $s\to -\mathrm{\infty }$

$\mathrm{\Phi }(z,s,a)=z^{-a}\mathrm{\Gamma }(1-s)\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}[2k\pi i-\log (z){]}^{s-1}{e}^{2k\pi ai}$

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

$\mathrm{\Phi }(-z,s,a)=z^{-a}\mathrm{\Gamma }(1-s)\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}[(2k+1)\pi i-\log (z){]}^{s-1}{e}^{(2k+1)\pi ai}$

for $|a|<1;\mathrm{\Re }(s)<0;z\notin (0,\mathrm{\infty }).$

An asymptotic series in the incomplete gamma function

$\mathrm{\Phi }(z,s,a)=\frac{1}{2a^s}+\frac{1}{z^a}\sum _{k=1}^{\mathrm{\infty }}\frac{e^{-2\pi i(k-1)a}\mathrm{\Gamma }(1-s,a(-2\pi i(k-1)-\log (z)))}{(-2\pi i(k-1)-\log (z){)}^{1-s}}+\frac{e^{2\pi ika}\mathrm{\Gamma }(1-s,a(2\pi ik-\log (z)))}{(2\pi ik-\log (z){)}^{1-s}}$

for $|a|<1;\mathrm{\Re }(s)<0.$

The representation as a generalized hypergeometric function is

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

Asymptotic expansion

The polylogarithm function ${\mathrm{L}\mathrm{i}}_n(z)$ is defined as

${\mathrm{L}\mathrm{i}}_0(z)=\frac{z}{1-z},{\mathrm{L}\mathrm{i}}_{-n}(z)=z\frac{d}{dz}{\mathrm{L}\mathrm{i}}_{1-n}(z).$

Let

${\mathrm{\Omega }}_a\equiv \{\begin{array}{ll}\mathbb{ℂ}\setminus [1,\mathrm{\infty })& \text{if }\mathrm{\Re }a>0,\\ z\in \mathbb{ℂ},|z|<1& \text{if }\mathrm{\Re }a\le 0.\end{array}$

For $|\mathrm{A}\mathrm{r}\mathrm{g}(a)|<\pi ,s\in \mathbb{ℂ}$ and $z\in {\mathrm{\Omega }}_a$, an asymptotic expansion of $\mathrm{\Phi }(z,s,a)$ for large $a$ and fixed $s$ and $z$ is given by

$\mathrm{\Phi }(z,s,a)=\frac{1}{1-z}\frac{1}{a^s}+\sum _{n=1}^{N-1}\frac{(-1{)}^n{\mathrm{L}\mathrm{i}}_{-n}(z)}{n!}\frac{(s{)}_n}{a^{n+s}}+O(a^{-N-s})$

for $N\in \mathbb{\mathbb{N}}$, 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 $N\in \mathbb{\mathbb{N}},\mathrm{\Re }a>1$ and $\mathrm{\Re }s>0$,

$\mathrm{\Phi }(z,s,a)-\frac{{\mathrm{L}\mathrm{i}}_s(z)}{z^a}=\sum _{n=0}^{N-1}{C}_n(z,a)\frac{(s{)}_n}{a^{n+s}}+O((\mathrm{\Re }a{)}^{1-N-s}+a{z}^{-\mathrm{\Re }a}),$

as $\mathrm{\Re }a\to \mathrm{\infty }$.

Software

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

References

• Apostol, T. M. (2010), "Lerch's Transcendent", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/25.14 .
• Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill, http://apps.nrbook.com/bateman/Vol1.pdf . (See § 1.11, "The function Ψ(z,s,v)", p. 27)
• "9.55." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press. 2015. ISBN 978-0-12-384933-5. 
• Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal 16 (3): 247–270, doi:10.1007/s11139-007-9102-0 . (Includes various basic identities in the introduction.)
• Jackson, M. (1950), "On Lerch's transcendent and the basic bilateral hypergeometric series ₂ψ₂", J. London Math. Soc. 25 (3): 189–196, doi:10.1112/jlms/s1-25.3.189 .
• Johansson, F.; Blagouchine, Ia. (2019), "Computing Stieltjes constants using complex integration", Mathematics of Computation 88 (318): 1829–1850, doi:10.1090/mcom/3401 .
• Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, ISBN 978-1-4020-1014-9 .

External links

• Aksenov, Sergej V.; Jentschura, Ulrich D. (2002), C and Mathematica Programs for Calculation of Lerch's Transcendent, http://aksenov.freeshell.org/lerchphi.html .
• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
• Garunkstis, Ramunas (2004). "Approximation of the Lerch Zeta Function". Lithuanian Mathematical Journal 44 (2): 140–144. doi:10.1023/B:LIMA.0000033779.41365.a5. http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf. 
• Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2015). "A generalization of Bochner's formula". https://hal.archives-ouvertes.fr/hal-02220916.  Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2004). "A generalization of Bochner's formula". Hardy-Ramanujan Journal 27. doi:10.46298/hrj.2004.150. 
• Weisstein, Eric W.. "Lerch Transcendent". http://mathworld.wolfram.com/LerchTranscendent.html. 
• Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds. (2010), "Lerch's Transcendent", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/25.14