Lerch zeta function explained

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( λ , s, \alpha) = \sum ^{inf ty (e ^{2\pi})/((n + \alpha) ^{s})} _{n =0} .

A related function, the Lerch transcendent, is given by

\Phi(z, s, \alpha) = \sum ^{inf ty (z ^{n})/((n + \alpha) ^{s})} _{n =0} .

The transcendent only converges for any real number

\alpha > 0 , where:|z| < 1 , orak{R}(s) > 1 , and |z| = 1 .^[1]

The two are related, as

\Phi(e ^{2\pi} , s,\alpha) = L( λ , s, \alpha).

Integral representations

The Lerch transcendent has an integral representation:

\Phi(z,s,a) = (1)/(\Gamma(s)) \int ^{inf ty (t ^{s-1} e ^{-at})/(1- ze ^{-t})} _{0} dt The proof is based on using the integral definition of the Gamma function to write\Phi(z,s,a)\Gamma(s) = \sum ^{inf ty} _{n =0} (z ^{n})/((n + a) ^{s}) \int ^{inf ty} _{0} x ^{s} e ^{-x} (dx)/(x) = \sum ^{inf ty} _{n =0} \int ^{inf ty} _{0} t ^{s} z ^{n} e ^{-(n+a)t} (dt)/(t) and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setminus [ 1 , inf ty), 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) =- (\Gamma( 1- s))/(2 \pi i) \int _{C} (( - 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) + 2 k\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) = (1)/(2 a ^{s}) + \int ^{inf ty} _{0} (z ^{t})/(dt + (a + t) ^{s}) (2)/(a ^{s-1}) \int ^{inf ty (\sin(s\arctan(t) - ta log (z)))/(( 1+ t ^{2)} ^{s/2} (e ^{2\pi} -1 ))} _{0} dt for\Re(a)> 0 \wedge |z|< 1 and\Phi(z,s,a) = (1)/(+ 2 a ^{s}) (log ^{s-1} ( 1 /z))/(\Gamma( 1- s,a log ( 1 /z)) + z ^{a}) (2)/(a ^{s-1}) \int ^{inf ty (\sin(s\arctan(t) - ta log (z)))/(( 1+ t ^{2)} ^{s/2} (e ^{2\pi} -1 ))} _{0} dt for\Re(a)> 0 .

Similar representations include

\Phi(z,s,a) = (1)/(2 a ^{s}) + \int ^{inf ty} _{0} (\cos(t log z)\sin ( s\arctan\tfrac{t)/(a) ) - \sin(t log z)\cos ( s\arctan\tfrac{t}{a} ) }{ ( a ^{2} + t ^{2)} ^{(s)/(2)} \tanh\pi t }dt,

and

\Phi( - z,s,a) = (1)/(2 a ^{s}) + \int ^{inf ty} _{0} (\cos(t log z)\sin ( s\arctan\tfrac{t)/(a) ) - \sin(t log z)\cos ( s\arctan\tfrac{t}{a} ) }{ ( a ^{2} + t ^{2)} ^{(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 ( \tfrac{\varphi}{ 2 \pi}, s, a ) = (1)/(a ^{s}) + (1)/(2 \Gamma(s)) \int ^{inf ty} _{0} (t ^{s-1} e ^{-at} ( e ^{i\varphi} - e ^{-t} ))/(\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 ^{inf ty} _{n =0} (1)/((n + \alpha) ^{s}) . The polylogarithm is another special case:rm{Li} _{s(z)=z\Phi(z,s,1)} = \sum ^{inf ty} _{n =1} (z ^{n})/(n ^{s}) . The Riemann zeta function is a special case of both of the above:\zeta(s) = \Phi( 1 ,s, 1 ) = \sum ^{inf ty} _{n =1} (1)/(n ^{s})

Other special cases include:

• The Dirichlet eta function:

η (s) = \Phi( -1 ,s, 1 ) = \sum ^{inf ty} _{n =1} (( -1 ) ^{n-1})/(n ^{s})

• The Dirichlet beta function:

\beta(s) = 2 ^{-s} \Phi( -1 ,s, 1 / 2 ) = \sum ^{inf ty} _{k =0} (( -1 ) ^{k})/(( 2 k +1 ) ^{s})

• The Legendre chi function:

\chi ^{- s} _{s(z) =2} z \Phi(z ^{2,s,1/2)} = \sum ^{inf ty} _{k =0} (z ^{2k+1})/(( 2 k +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( λ , s, \alpha) may be expressed as a finite sum over the Hurwitz zeta function. Suppose $\backslash lambda\; =\; \backslash frac$ with p, q \in \Z and q > 0 . Then z = \omega = e ^{2 \pi i (p)/(q)} and \omega ^{q} = 1 .\Phi(\omega, s, \alpha) = \sum ^{inf ty (\omega ^{n})/((n + \alpha) ^{s})} _{n =0} = \sum ^{q -1} _{m =0} \sum ^{inf ty} _{n =0} (\omega ^{qn})/((qn + m + \alpha) ^{s}) = \sum ^{q -1} _{m =0} \omega ^{m} q ^{-s} \zeta \left( s, (m + \alpha)/(q) \right)

Various identities include:

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

and

\Phi(z,s -1 ,a) = \left(a + z (\partial)/(\partial z) \right) \Phi(z,s,a)

and

\Phi(z,s +1 ,a) =- (1)/(s) (\partial)/(\partial a) \Phi(z,s,a).

Series representations

A series representation for the Lerch transcendent is given by

\Phi(z,s,q) = (1)/(1- z) \sum ^{inf ty} _{n =0} \left( (- z)/(1- z) \right) ^{n} _{k =0} ( -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)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.^[2]

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for^[3]

\left| log (z)\right| < 2 \pi;s ≠ 1 , 2 , 3 , ... ; a ≠ 0 , -1 , -2 , ... \Phi(z,s,a) = z ^{-a} \left[\Gamma( 1- s)\left( -log (z)\right) ^{s-1} + \sum ^{inf ty} _{k =0} \zeta(s - k,a) (log ^{k} (z))/(k!) \right]

If n is a positive integer, then

\Phi(z,n,a) = z ^{-a} \left\{ \sum _{{k=0} \atop k ≠ n -1 } ^{infty} \zeta(n - k,a) (log ^{k} (z))/(+ \left[\psi(n) - \psi(a) -log ( -log (z))\right] k!) (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 ^{inf ty} _{k =0} \Phi(z,s + k,a)(s) _{k} (( - 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 ^{n} _{k =0} (z ^{k})/((a + k) ^{s}) + z ^{inf ty} _{m =0} ( 1- m - s) _{m} \operatorname{Li} _{s+m} (z) ((a + n) ^{m})/(m!) ; a → - n

A special case for n = 0 has the following series

\Phi(z,s,a) = (1)/(a ^{s}) + \sum ^{inf ty} _{m =0} ( 1- m - s) _{m} \operatorname{Li} _{s+m} (z) (a ^{m})/(m!) ; |a|< 1 , where \operatorname{Li} _{s(z)} is the polylogarithm.

An asymptotic series for

s → -inf ty \Phi(z,s,a) = z ^{-a} \Gamma( 1- s)\sum ^{inf ty} _{k =-inf ty} [ 2 k\pi i -log (z)] ^{s-1} e ^{2k\pi} for |a|< 1 ;\Re(s)< 0 ;z\notin ( -inf ty, 0 ) and\Phi( - z,s,a) = z ^{-a} \Gamma( 1- s)\sum ^{inf ty [( 2 k +1 )\pi} _{k =-inf ty} i -log (z)] ^{s-1} e ^{(2k+1)\pi} for |a|< 1 ;\Re(s)< 0 ;z\notin ( 0 , inf ty).

An asymptotic series in the incomplete gamma function

\Phi(z,s,a) = (1)/(+ 2 a ^{s}) (1)/(z ^{a}) \sum ^{inf ty (e ^{-2\pi} \Gamma( 1- s,a( -2 \pi i(k -1 ) -log (z))))/(( -2 \pi i(k -1 ) -log (z)) ^{1-s})} _{k =1} + (e ^{2\pi} \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^[4]

\Phi(z,s,\alpha) = (1)/(\alpha ^{s}) {} _{s+1} F _{s\left(\begin{array}{c} 1,\alpha,\alpha,\alpha, … \\ 1+\alpha,1+\alpha,1+\alpha, … \\ \end{array}\mid} z\right).

Asymptotic expansion

The polylogarithm function

Li _{n(z)} is defined asLi _{0 (z) = (z)/(1- z)} , Li _{-n} (z) = z (d)/(dz) Li _{1-n} (z). Let\Omega _{a} \equiv\begin{cases} C \setminus[ 1 , inf ty) & if \Re a > 0 , \\ {z \in C , |z|< 1 } & if \Re a \le 0 . \end{cases} For | Arg (a)|<\pi, s \in 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) = (1)/(1- z) (1)/(a ^{s}) + \sum ^{N -1} _{n =1} (( -1 ) ^{n} Li _{-n} (z))/(n!) ((s) _{n})/(a ^{n+s}) + O(a ^{-N-s} ) for N \in N , where (s) _{n} = s (s +1 ) … (s + n -1 ) is the Pochhammer symbol.^[5]

Let

f(z,x,a) \equiv (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 N , \Re a > 1 and \Re s > 0 ,\Phi(z,s,a) - (Li _{s} (z))/(z ^{a}) = \sum ^{N -1} _{n =0} C _{n} (z,a) ((s) _{n})/(a ^{n+s}) + O\left( (\Re a) ^{1-N-s} + a z ^{-\Re} \right), as \Re a \to inf ty .^[6]

Software

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

References

• .
• . (See § 1.11, "The function Ψ(z,s,v)", p. 27)
• Book: Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . Victor Hugo . Moll . Victor Hugo Moll . Scripta Technica, Inc. . Table of Integrals, Series, and Products . Academic Press . 2015 . October 2014 . 8 . English . 978-0-12-384933-5 . 2014010276 . 2016-02-21-->. Gradshteyn and Ryzhik . 9.55..
• . (Includes various basic identities in the introduction.)
• .
• .
• .

External links

• .
• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
• Ramunas. Garunkstis. Approximation of the Lerch Zeta Function. 2004. Lithuanian Mathematical Journal. 44. 2. 140–144. 10.1023/B:LIMA.0000033779.41365.a5. 123059665 .
• Web site: S.. Kanemitsu. Y.. Tanigawa. H.. Tsukada. A generalization of Bochner's formula. 2015. S.. Kanemitsu. Y.. Tanigawa. H.. Tsukada. 10.46298/hrj.2004.150. Hardy-Ramanujan Journal. A generalization of Bochner's formula. 2004. 27. free.

Notes and References

1. https://arxiv.org/pdf/math/0506319.pdf
2. Web site: The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function . 27 April 2020 . 28 April 2020.
3. B. R. Johnson . Generalized Lerch zeta function . Pacific J. Math. . 53 . 1 . 1974 . 189–193 . 10.2140/pjm.1974.53.189 . free.
4. J. E.. Gottschalk. E. N. . Maslen. Reduction formulae for generalized hypergeometric functions of one variable. J. Phys. A . 1988. 21. 9. 1983–1998. 10.1088/0305-4470/21/9/015. 1988JPhA...21.1983G.
5. Ferreira . Chelo . López . José L. . Asymptotic expansions of the Hurwitz–Lerch zeta function . Journal of Mathematical Analysis and Applications . October 2004 . 298 . 1 . 210–224 . 10.1016/j.jmaa.2004.05.040. free .
6. Cai . Xing Shi . López . José L. . A note on the asymptotic expansion of the Lerch's transcendent . Integral Transforms and Special Functions . 10 June 2019 . 30 . 10 . 844–855 . 10.1080/10652469.2019.1627530. 1806.01122 . 119619877 .