The Lerch zeta function (after Mathias Lerch ) is a very general zeta function . Very many series of reciprocal powers (including the Hurwitz zeta function and the polylogarithm ) can be represented as a special case of this function.
definition
The two functions
![{\ displaystyle L (\ lambda, \ alpha, s) = \ sum _ {n = 0} ^ {\ infty} {\ frac {\ exp (2 \, \ pi \, i \, \ lambda \, n) } {(n + \ alpha) ^ {s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8b83c91d1c24031ee2fa86f6bd074eb47c7626)
and
![{\ displaystyle \ Phi (z, s, \ alpha) = \ sum _ {n = 0} ^ {\ infty} {\ frac {z ^ {n}} {(n + \ alpha) ^ {s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce334a553e1d950a0235b2aa2290236bbdf3ed4)
are called Lerch's zeta function . The relationship between the two is through
![{\ displaystyle \, \ Phi (\ exp (2 \, \ pi \, \, i \, \ lambda), s, \ alpha) = L (\ lambda, \ alpha, s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6afbc75ac4a48eb046bb092d50aebb45cb4f81)
given.
Special cases and special values
![{\ displaystyle \, \ zeta (s, n) = L (0, n, s) = \ Phi (1, s, n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757b4cb311ea82074ef25393967155c9ea9eb808)
![{\ displaystyle \, {\ textrm {Li}} _ {s} (z) = z \, \ Phi (z, s, 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d4d072337cba1aea66389b7078d975a1215a5e)
![{\ displaystyle \, \ chi _ {n} (z) = 2 ^ {- n} \, z \, \ Phi (z ^ {2}, n, {\ tfrac {1} {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebcecbf0b67351d59d976f55971b5c432587b03)
![{\ displaystyle \, \ zeta (s) = \ Phi (1, s, 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec7bf617e150b9733cb68d8db1641b34e7430ae)
![{\ displaystyle \, \ eta (s) = \ Phi (-1, s, 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52490a89d2d35e5eda7d413c61fc9a8db626f359)
![{\ displaystyle \ beta (s) = 2 ^ {- s} \, \ Phi (-1, s, {\ tfrac {1} {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6702f88fedecf2e84c0ad4d781eba0ea75e556c1)
The following special cases also apply (selection):
![{\ displaystyle \ Phi (z, s, 1) = {\ frac {\ mathrm {Li} _ {s} (z)} {z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c137b82dfac8ffe2d821461b418de2dce6d9032)
![{\ displaystyle \ Phi (z, 0, a) = {\ frac {1} {1-z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac2389c0c3e19ce72df9629adbc6a16e2c494e0)
![{\ displaystyle \ Phi (0, s, a) = \ left (a ^ {2} \ right) ^ {- {\ frac {s} {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5658adb6e7ab8c9a6d0f04972b50e2669486e3)
![{\ displaystyle \ Phi (0, s, a) = a ^ {- s} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1626de5c322f1f037c82d696921c379aff874d58)
![{\ displaystyle \ Phi (z, 1,1) = - {\ frac {\ log (1-z)} {z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3628ca0b692ea82c07479667b3c8a6c5834f4db1)
![{\ displaystyle \ Phi (1, s, {\ tfrac {1} {2}}) = (2 ^ {s} -1) \ zeta (s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e73826f7f7b75d5ced9628042bd7956ef659cbc6)
![{\ displaystyle \ Phi (-1, s, 1) = (1-2 ^ {1-s}) \ zeta (s) \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e32ba89a7bfb7f32f992a6e47572ef0fd8c0632)
![{\ displaystyle \ Phi (0,1, a) = {\ frac {1} {\ sqrt {a ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4092e6d8070c3971d0d1ad516ccd85e91d1be54d)
Furthermore is
![{\ displaystyle {\ begin {aligned} & \ Phi (-1,2, {\ tfrac {1} {2}}) & = & \; 4 \, G \\ & {\ frac {\ partial \ Phi} {\ partial s}} (- 1, -1,1) & = & \; \ log \ left ({\ frac {A ^ {3}} {{\ sqrt [{3}] {2}} \, {\ sqrt [{4}] {\ mathrm {e}}}}} \ right) \\ & {\ frac {\ partial \ Phi} {\ partial s}} (- 1, -2,1) & = & \; {\ frac {7 \, \ zeta (3)} {4 \, \ pi ^ {2}}} \\ & {\ frac {\ partial \ Phi} {\ partial s}} (- 1, -1, {\ tfrac {1} {2}}) & = & \; {\ frac {G} {\ pi}} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/604103b7e47a2180ac4304d0b2d034077e64001f)
with the Catalan constant
, the Glaisher-Kinkelin constant
and the Apéry constant of
the Riemann zeta function.
More formulas
Integral representations
One possible integral representation is
-
For
The curve integral
![{\ displaystyle \ Phi (z, s, a) = - {\ frac {\ Gamma (1-s)} {2 \, \ pi \, i}} \ int \ limits _ {0} ^ {\ infty} {\ frac {(-t) ^ {s-1} \ mathrm {e} ^ {- a \, t}} {1-z \, \ mathrm {e} ^ {- t}}} \, \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/146c75ac94912207dfeed110a9207b070f52cc78)
with must not contain the points .
![{\ displaystyle \ mathrm {Re} \; a> 0, \; \ mathrm {Re} \; s <0, \; z <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b6f14777ac91992be9af082c38762bdc7d423f)
![{\ displaystyle t = \ log z + 2 \, k \, \ pi \, i, \; k \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7b20bccb979aec00eca87c25c551e40a6a0223)
Furthermore is
![{\ displaystyle \ Phi (z, s, a) = {\ frac {1} {2 \, a ^ {s}}} + \ int \ limits _ {0} ^ {\ infty} {\ frac {z ^ {t}} {(a + t) ^ {s}}} \, \ mathrm {d} t + {\ frac {2} {a ^ {s-1}}} \ int \ limits _ {0} ^ { \ infty} {\ frac {\ sin (s \ arctan ta \, t \ log z)} {(1 + t ^ {2}) ^ {s / 2} \ cdot (\ mathrm {e} ^ {2 \ , \ pi \, a \, t} -1)}} \, \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2a3176ebe513624adf0113539f439d153991cc)
for and .
![{\ displaystyle \ mathrm {Re} \; a> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0845680d95c8a52c81984ffb28ecf977aa266ef)
![| z | <1](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c0fa57b899b653a3823f85f43fd666309c09b3)
Likewise is
![{\ displaystyle \ Phi (z, s, a) = {\ frac {1} {2 \, a ^ {s}}} + {\ frac {\ log ^ {s-1} {\ dfrac {1} { z}}} {z ^ {a}}} \, \ Gamma (1-s, a \ log {\ dfrac {1} {z}}) + {\ frac {2} {a ^ {s-1} }} \ int \ limits _ {0} ^ {\ infty} {\ frac {\ sin (s \ arctan ta \, t \ log z)} {(1 + t ^ {2}) ^ {s / 2} \ cdot (\ mathrm {e} ^ {2 \, \ pi \, a \, t} -1)}} \, \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea76e6c6e67246d797efdb141219a07119d6f7d)
for .
![{\ displaystyle \ mathrm {Re} \, a> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d16ab20b609969885dba00575fe889644e56493)
Series representations
A series representation for the transcendent lark is
![{\ displaystyle \ 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} {n \ choose k} (q + k) ^ {- s}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6313c18504dc30909aedec8b3a86d16d83c145c)
It applies to all and complex with ; compare the series representation of the Hurwitz zeta function .
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle \ mathrm {Re} \, z <{\ tfrac {1} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b221ca9db4dbcd55b37bda7dd36ab98dea44638)
If is positive and whole, then applies
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![{\ displaystyle \ Phi (z, n, a) = z ^ {- a} \ left \ {\ sum _ {{k = 0} \ atop k \ neq n-1} ^ {\ infty} \ zeta (nk , a) {\ frac {\ log ^ {k} z} {k!}} + \ left [\ Psi (n) - \ Psi (a) - \ log (- \ log z) \ right] {\ frac {\ log ^ {n-1} z} {(n-1)!}} \ right \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b54a52635ff1519524a9ae99a5060e01582e8f1)
A Taylor series of the third variable is through
![{\ displaystyle \ Phi (z, s, a + x) = \ sum _ {k = 0} ^ {\ infty} \ Phi (z, s + k, a) (s, k) {\ frac {(- x) ^ {k}} {k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0180fc818c66843043ded164591974bbc1ba577c)
given for using the Pochhammer symbol .
![{\ displaystyle (s, k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f99142989ff1432e322c9c8a0277f33dc4ad2e)
The following applies in the
limit value![{\ displaystyle a \ rightarrow -n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e24c8fb1b00a3ca095edc4b813decc978c4f1432)
-
.
The special case has the following series:
![n = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae)
![{\ displaystyle \ Phi (z, s, a) = {\ frac {1} {a ^ {s}}} + \ sum _ {m = 0} ^ {\ infty} (1-ms, m) \, \ mathrm {Li} _ {s + m} (z) {\ frac {a ^ {m}} {m!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4d81898a4d5c14cc01f2dd2225293912a8b1e7)
for .
![{\ displaystyle | a | <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1dfd008c81cab0fbba6665b2381298668aca74)
The asymptotic expansion for is given by
![{\ displaystyle s \ rightarrow - \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8f4fc5f1a768c4b7143c1bb863084eff4eea60)
![{\ displaystyle \ Phi (z, s, a) = z ^ {- a} \, \ Gamma (1-s) \ sum _ {k = - \ infty} ^ {\ infty} \ left [2 \, k \, \ pi \, i- \ log z \ right] ^ {s-1} \ mathrm {e} ^ {2 \, k \, \ pi \, a \, i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3db0d7fc935e69ada0183b6c210d447063a525)
for and
![{\ displaystyle | a | <1, \; \ mathrm {Re} \; s <0, \; z \ notin (- \ infty, 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e347a28edc7b6d296e874a6b9d548a267bb7ce9)
![{\ displaystyle \ Phi (-z, s, a) = z ^ {- a} \, \ Gamma (1-s) \ sum _ {k = - \ infty} ^ {\ infty} \ left [(2 \ , k + 1) \ pi \, i- \ log z \ right] ^ {s-1} \ mathrm {e} ^ {(2 \, k + 1) \ pi \, a \, i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b2d27d83c654da931d25e9d408a9edc0617f42)
if .
![{\ displaystyle | a | <1, \; \ mathrm {Re} \, s <0, \; z \ notin (0, \ infty)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb58d3974cdc3945275b0f927fd566c3eb78b24)
Using the incomplete gamma function holds
![{\ displaystyle \ Phi (z, s, a) = {\ frac {1} {2 \, a ^ {s}}} + {\ frac {1} {z ^ {a}}} \ sum _ {k = 1} ^ {\ infty} {\ frac {\ mathrm {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 {\ mathrm {e} ^ {2 \, \ pi \, i \, k \, a} \, \ Gamma (1-s, a \, (2 \, \ pi \ , i \, k- \ log z))} {(2 \, \ pi \, i \, k- \ log z) ^ {1-s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d9b99600994857bef01ee17a4a7ce89595fc6d)
with and .
![{\ displaystyle | a | <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1dfd008c81cab0fbba6665b2381298668aca74)
![{\ displaystyle \ mathrm {Re} \, s <0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3696df18781e99c3e858bacda1fddb4ea6749d86)
Identities and other formulas
![{\ displaystyle \ Phi (z, s, a) = z ^ {n} \, \ Phi (z, s, a + n) + \ sum _ {k = 0} ^ {n-1} {\ frac { z ^ {k}} {(k + a) ^ {s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93319ac7fca570bcd3b5ac8b649c0cde4060c7e8)
![{\ displaystyle \ Phi (z, s-1, a) = \ left (a + z {\ frac {\ partial} {\ partial z}} \ right) \ Phi (z, s, a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8e616ebacd2f4d44c5455d1df281fc708d4a66)
![{\ displaystyle \ Phi (z, s + 1, a) = - \, {\ frac {1} {s}} \, {\ frac {\ partial} {\ partial a}} \ Phi (z, s, a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f06f9910274a73c0a38baf0120d1f5a1e25b436)
Furthermore applies to the integral representation with or
![{\ displaystyle \ {z \ in \ mathbb {C} \, \ setminus [1, \ infty) {\ text {and}} \ mathrm {Re} \; s> -2 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8f353eaaa357c85687b99822fe0a019530e1de)
![{\ displaystyle \ int \ limits _ {0} ^ {1} \ int \ limits _ {0} ^ {1} {\ frac {x ^ {u-1} \ cdot y ^ {v-1}} {1 -x \, y \, z}} (- \ log (x \, y)) ^ {s} \, \ mathrm {d} x \, \ mathrm {d} y = \ Gamma (s + 1) \ , {\ frac {\ Phi (z, s + 1, v) - \ Phi (z, s + 1, u)} {uv}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ede3bc449e3cdf8a93d14dff58e45b6bfb177658)
and
-
.
literature
-
Mathias Lerch : Démonstration élémentaire de la formule :, L'Enseignement Mathématique 5 (1903): pp. 450–453
![{\ displaystyle \ textstyle {\ frac {\ pi ^ {2}} {\ sin ^ {2} {\ pi x}}} = \ sum _ {\ nu = - \ infty} ^ {\ infty} {\ frac {1} {(x + \ nu) ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90cc1e0589a4c5a8f042b86887eacec068665071)
- M. Jackson: On Lerch's transcendent and the basic bilateral hypergeometric series
, J. London Math. Soc. 25 (3), 1950: pp. 189-196
- Jesús Guillera, Jonathan Sondow: Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . In: Ramanujan J. Volume 16, Number 3, 2008, pages 247-270; see. in arxiv
- Antanas Laurinčikas and Ramūnas Garunkštis: The Lerch zeta-function , Dordrecht: Kluwer Academic Publishers, 2002, ISBN 978-1-4020-1014-9 online
Web links
Individual evidence
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/03/ShowAll.html
-
↑ Guillera, Sondow 2008, Theorem 3.1 (see Ref.)