Kummer s Formula for Multiple Gamma Functions

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1 Kummer s Formula for Multiple Gamma Fuctios Shi-ya Koyama Keio Uiversity Nobushige Kurokawa Tokyo Istitute of Techology Ruig title Kummer s Formula Abstract We geeralize Kummer s formula o a expressio of the gamma fuctio to that of multiple gamma fuctios usig the fuctioal equatio for multiple Hurwitz zeta fuctios 000 Mathematics Subject Classificatio: 11M06 1 Itroductio Kummer s formula for the usual gamma fuctio Γx is the followig idetity: log Γx = 1 π log siπx + logπ + γ π siπx + 1 cosπx + 1 logπ 11 for 0 < x < 1, which was discovered by Kummer [9] 187 A spectacular applicatio is give by lettig x be a ratioal umber: we have formulas for special values of Dirichlet L-fuctios usig the gamma fuctio For example, whe x = 1/ we get L 1, χ = π log Γ due to Malmsté [0] 189, where 1 Ls, χ = :odd + π γ + 3 log π + log 1 1 s is the Dirichlet L-fuctio for the o-trivial character χ modulo This result is equivaletly writte as 1 L 0, χ = log Γ 3 log log π Preseted at the coferece o Zetas ad Trace Formulas i Okiawa, November 00 1

2 via the fuctioal equatio We remark that L0, χ = 1/ ad L1, χ = π/, so Malmsté s result gives the difficult secod term of the Taylor expasio for Ls, χ at s = 0 ad s = 1 Marmsté s study motivated the followig excellet formula of Lerch [18] 189, where log Γx = ζ 0, x + 1 logπ 1 ζs, x = + x s =0 is the Hurwitz zeta fuctio [8] ad the differetiatio is i the first variable s Combiig this with Kroecker s limit formula Lerch [19] 1897 obtaied the formula λ=1 D λ log Γ = h log h log π log a + λ 3 3 a,b,c a,b,c logϑ 10 αϑ 10 β, 13 where ϑ 1 is the usual otatio of theta fuctio so ϑ 1 is essetially η 3, Q D is a imagiary quadratic field of discrimiat D ad class umber h, a, b, c rus over h quadratic forms correspodig to the ideal classes, ad = D It would be coveiet to look at the formula 163 of Ladau [17] I recet years this formula is called Chowla-Selberg formula after the 70 years late paper of Chowla-Selberg [] 1967 with o metio to Lerch [19] ufortuately Cocerig the history aroud Kummer s formula 11 ad the above two Lerch s formulae 1, 13 we refer to the excellet survey [17] writte by Ladau, which is the famous first paper o the prime ideal theorem geeralizig the usual prime umber theorem to algebraic umber fields The purpose of this paper is to preset a geeralizatio of Kummer s formula i the case of the multiple gamma fuctio Γ r x This multiple gamma fuctio origiates from Bares [1] ad it is defied as Γ r x = expζ r0, x where ζ r s, x = = 1,, r = r + x s + r 1 + x s r 1 =0 is the multiple Hurwitz zeta fuctio We have may applicatios of Γ r x i Sarak [3], Voros [5], Shitai [] ad our previous papers [10] - [15] From Γ 1 x = Γx/ π, we

3 ca write Kummer s formula as follows: log Γ 1 x = ζ 0, x = 1 log siπx π + logπ + γ siπx π Our result is a geeralizatio of this formula: Theorem 1 Let 0 < x < 1 The we have 1 log Γ x = 1 π log Γ 3 x = 1 π log cosπx logπ + γ 1 π + 1 π cosπx 1 siπx log siπx logπ + γ 3 3 8π 3 1 8π cosπx x cosπx log Γ x + 1 x log Γ 1 x siπx 3 x 1 log Γ 1 x I the text we show some applicatios We otice that our method is quite simple to obtai ζ r0, x usig the fuctioal equatio ζ r s, x = ξ r r s, x where ξ r s, x is a Dirichlet series itimately related to ζ r s, x: log Γ r x = ξ rr, x This atural method was discovered by Hardy [] to reprove the origial Kummer s formula I [5, 6], Hardy studied the case of the double gamma fuctio, but he did ot obtai the geeralizatio of Kummer s formula Ackowledgemet This paper was preseted at the symposium Zetas ad Trace Formulas i Okiawa from November 7 to 9 of 00 The authors would like to express their hearty thaks to all the participats of Okiawa symposium We also thak Lagua Garde Hotel ad Okiawa Covesio Ceter for supplyig the excellet atmosphere with blue sky ad sea 3

4 Multiple Gamma Fuctios We prepare o the multiple gamma fuctio Γ r x We refer to Bares[1] for the geeral theory First we kow that the fuctio Γ r x is a meromorphic fuctio of order r Γ r x 1 beig a etire fuctio of order r, ad it has a simple pole at x = 0: Γ r x 1 ρ r x as x 0 We remark that Bares [1] used Γ B r x ormalized as Γ B r x = ρ r Γ r x 1 x as x 0 For later use we calculate ρ r for r = 1,, 3 here Theorem 1 ρ 1 = π ρ = πe ζ 1 = exp ζ π + 7 logπ + γ Proof From we have ρ 3 = πe 3 ζ 1 1 ζ3 ζ = exp 8π 3ζ + 5 logπ + γ 1 π 8 Hece lettig x 0 we get ζ r s, x + 1 = ζ r s, x ζ r 1 s, x Γ r x + 1 = Γ r xγ r 1 x 1 Γ r 1 = ρ r 1 ρ r Here we uderstad that ζ 0 s, x = x s, Γ 0 x = x 1 ad ρ 0 = 1 The first result ρ 1 = π follows from Lerch s formula Γ 1 x = Γx π 1 πx as x 0 Now we calculate ad ρ = ρ 1 Γ 1 1 = πγ 1 1 ρ 3 = ρ Γ = πγ 1 1 Γ 3 1 1

5 Sice ζ s, x = = = x s =0 + x + 1 x + x s =0 + x s x + x s =0 = ζs 1, x + 1 xζs, x =0 ad we obtai ζ 3 s, x = + x s =0 = 1 + x + 3 x + x + x 1x + x s = 1 =0 3 x ζs, x + ζs 1, x + ζ 0, x = ζ 1, x + 1 xζ 0, x x 1x ζs, x, ad ζ 30, x = 1 ζ, x + 3 x ζ x 1x 1, x + ζ 0, x Hece, lettig x = 1 ad remarkig ζs, 1 = ζs, we have Γ 1 = expζ 0, 1 = expζ 1 ad Thus we have ad 1 Γ 3 1 = expζ 30, 1 = exp ζ + 1 ζ 1 ρ = π exp ζ 1 ρ 3 = π exp 3 ζ 1 1 ζ Now we use the fuctioal equatio for ζs: ζ1 s = π s Γs cos 5 πs ζs

6 By differetiatig both sides of this fuctioal equatio at s =, we have ζ 1 = π Γcos πζ + π logπγcos πζ +π Γ cos πζ = ζ π + logπ 1 γ 1 1, where we used ζ = π /6 ad Γ = 1 γ Hece ρ = π exp = exp ζ π + logπ 1 ζ π + 7 logπ + γ 1 1 Next, from the fuctioal equatio for ζs we have ζ = π 3 Γ3 π si 3π 1 γ 1 ζ3 Hece Thus logπ ρ 3 = exp = exp ζ = ζ3 π ζ π logπ + γ ζ3 8π 3ζ + 5 logπ + γ 1 π 8 1 ζ3 π 3 Geeralized Kummer s Formula To prove Theorem 1 we show Theorem 3 Let k 1 be a iteger 1 Whe k is odd, ζ k, x = k+1 1 k! π k+1 + { log cosπx logπ + γ k k 6 cosπx k+1 π } siπx k+1

7 Whe k is eve, { ζ k, x = 1 k k! log siπx π k+1 k+1 + logπ + γ k siπx k+1 + π } cosπx k+1 Proof We have the followig fuctioal equatio for ζs, x proved by Hurwitz [8]: πs ζs, x = π s 1 cosπx πs siπx Γ1 s si + cos 1 s 1 s Hece we obtai ζ k, x = π k 1 logπ k! Γ 1 + k πk cosπx si + cos k+1 πk +π k 1 k! si + cos πk + π cos πk + π si πk log siπx k+1 cosπx k+1 siπx k+1 πk log cosπx k+1 Thus, usig Γ Γ k + 1 = k γ we get 1 ad Proof of Theorem 1 From the proof of Theorem we see the relatios siπx k+1 log Γ x = ζ 0, x = ζ 1, x + 1 xζ 0, x = ζ 1, x + 1 x log Γ 1 x 7

8 ad log Γ 3 x = ζ 30, x = 1 ζ, x + 3 x ζ x 1x 1, x + ζ 0, x = 1 3 ζ, x + x ζ 0, x 1 x ζ 0, x = 1 3 ζ, x + x x 1 log Γ x log Γ 1 x Hece we have Theorem 1 from Theorem 3 Applicatios As the first applicatio of our geeralizatio of Kummer s formula we calculate some special values Γ r x for r = ad 3 Before this we explai our method i the simple case r = 1: lettig x = 1/ i Kummer s formula 1 for Γ 1 x we have Γ 1 = exp = exp 1 log = 1 ad Γ B = ρ 1 Γ 1 = 1 π = π Actually i this case r = 1 these results are also direct from Γ 1 x = Γx/ π ad Γ B 1 x = Γx usig the well-kow Γ1/ = π Theorem 1 Γ 1 = exp ζ π + log π + γ 1 log Γ B 1 = exp 3ζ + 5 log π + γ 1 + log π 8 3 8

9 3 Γ 3 1 3ζ3 = exp 3π ζ π + log π + γ 1 3 log 16 Γ B 3 1 7ζ3 = exp 3π ζ + log π + γ log π 6 16 Proof 1 Let x = 1/ i Theorem 11 The we have log Γ 1 We otice = 1 π log cosπ logπ + γ 1 π cosπ s = 1 1 s = 1 s ζs = 1 s 1ζs cosπ ad its differetiatio log cosπ s = log 1 s ζs 1 s 1ζ s Hece ad Thus log Γ 1 cosπ = 1 ζ = π 1 log cosπ = log ζ + 1 ζ = π log 1 + ζ = log ζ π + logπ + γ 1 log = ζ π + log π + γ 1 log log Γ 1 1

10 This gives 1 Sice Γ B 1/ = ρ Γ 1/, follows from 1 ad the formula for ρ i Theorem 3 Set x = 1/ i Theorem 1 The where we used log Γ 3 1 = 1 8π = 3ζ3 3π + log Γ cosπ 3 cosπ + log Γ log 16, log Γ 1 = 1 3 1ζ3 = 3 ζ3 1 ad Γ 1 1/ = 1/ Hece we obtai 3 from 1 Sice Γ B 3 1/ = ρ 3 Γ 3 1/, follows from 3 ad the fomula for ρ 3 i Theorem Values of Γ r x at ratioal umbers are also expressed via special values of Dirichlet L-fuctios besides ζs through the geeralized Kummer s formula Here we report the followig typical example Theorem 5 1 Γ L, χ = exp 3L 1, χ π π ζ 19 log π + 19γ + log 1 +, 16π 96 where Ls, χ is the Dirichlet L-fuctio for the o-trivial character χ modulo Proof Let x = 1/ i Theorem 11 The we have log Γ 1 We otice = 1 π cos π s = log cos π logπ + γ 1 π 1 s + 1 π cos π si π + 3 log Γ 1 = s 1 s 1ζs = 1 s s ζs ad its differetiatio log cos π = 1 s s ζ s + s s log ζs s 10 1

11 Hece ad Usig we have cos π si π s = = ζ 8 = π 8 log cos π = ζ 8 :odd 1 1 = Ls, χ s 1 Γ Thus we obtai Theorem 5 from the result of Malmsté [0] 1 log Γ 1 = L 1, χ π = ζ 16π + logπ + γ 1 + L, χ π log Γ 1 + log π + γ, 1 which is obtaied by settig x = 1/ i Kummer s formula 1: 1 log Γ 1 = 1 log si π π + logπ + γ si π + 1 cos π π = 1 π L 1, χ + logπ + γ L1, χ π 1 = 1 π L 1, χ + log π + γ, 3 where we used L1, χ = π/ ad 1 1 = log Now let S r x = Γ r x 1 Γ r r x 1r be the multiple sie fuctio studied i [13] As the secod applicatio of the geeralized Kummer s formula we prove basic properties of S r x for r = ad 3 This geeralizes the usual sie fuctio: We otice that S 1 x = Γ 1 x 1 Γ 1 1 x 1 = S 1 x = exp π ΓxΓ1 x = siπx for 0 < x < 1 from Kummer s formula for Γ 1 x 11 cosπx

12 Theorem 6 1 S x = exp 1 siπx + x 1 π = exp 1 siπx S π 1 x 1 x for 0 < x < 1 cosπx 3 S S x = πx 1 cotπx 1 S = 5 Proof Sice ad we have S x = πxe x S x = exp π Hece, from 1 of Theorem 1 we have log S x = 1 π ad 1 + x 1+ 1 x 1 e x x 1 S x = Γ x 1 Γ x 0 t cotπtdt Γ x = Γ 1 xγ 1 1 x 1, S x = Γ x 1 Γ 1 xγ 1 1 x 1 siπx S S x = x 1 S 1 S 1 x = x 1 S 1 S 1 x = x 1π cotπx, 1 x 1 log S 1 x cosπx log S 1 x

13 where we used log S 1 x = log Γ 1 x log Γ 1 1 x = cosπx Thus 1 ad are proved We have 3 by settig x = 1/ i 1 For the proof of let S z = e z followig Hölder [7] The we easily obtai 1 z 1 + z e z S S x = πx cotπx ad S 1/ = [7, 13] Hece it is sufficiet to show that S x = S x 1 S 1 x We kow S 1/ = ad S 1 1/ = Thus the both sides are at x = 1/, ad the differetiatios of the both sides tur out to be x 1 cotπx Hece we get Lastly 5 follows from sice both sides are 1 at x = 1 Remark 1 These results have further applicatios cotaiig expressios of the difficult special values of Dirichlet L-fuctios [13] Especially 5 gives the calculatio of the gamma factor of the Selberg zeta fuctio for a Riema surface Sarak [3] ad Voros [5] The situatio is similar i the higher dimesioal case, where we use the multiple sie fuctio S r x with r beig the dimesio [13] Theorem S 3 x = exp π 1 = exp π for 0 < x < 1 cosπx x 3 3 π cosπx 3 S x 3 x S 1 x x 1 siπx x 1x cosπx S 3 x = π x 1x cotπx S 3 13

14 3 S 3 1 = 38 exp 3 16π ζ3 S 3 x = πe ζ xe x 3 x Proof Usig Γ r x = Γ r x 1Γ r 1 x 1 1 we have S 3 x = Γ 3 x 1 Γ 3 3 x 1 = Γ 3 x 1 Γ 3 xγ x 1 1 Hece, from Theorem 1, for 0 < x < x x 1 e x 3x = Γ 3 x 1 Γ 3 1 xγ 1 x 1 Γ 1 x 1 Γ 1 1 x 1 = Γ 3 x 1 Γ 3 1 x 1 Γ 1 x Γ 1 1 x 1 log S 3 x = 1 cosπx 3 π 3 x x 1 log Γ x + log Γ 1 x 1 + x log Γ 1 x + x log Γ 11 x + log Γ 1 x log Γ 1 1 x = 1 cosπx 3 + π 3 x logγ 1 xγ x 1 x 1 x + log Γ 1 x + 1 log Γ 1 1 x = 1 cosπx 3 + π 3 x logs xγ 1 1 x x 1 x + log Γ 1 x + 1 log Γ 1 1 x = 1 cosπx 3 + π 3 x x 1 log S x log S 1 x, where we used that ad Γ 1 xγ x 1 = S xγ 1 1 x Γ 1 x 1 Γ 1 1 x 1 = S 1 x 1

15 From 1 we have S 3 x = 1 siπx S 3 π 3 S + x x S log S x + x 1 log S 1 x x 1 S 1 x S 1 Hece, usig the previous formulas for log S r x ad S rx/s r x r = 1,, we obtai S 3 x = 1 S 3 π +π siπx + 1 π x 1 x 3 = π x 1x cotπx siπx x 1 cotπx 3 From 1 1 S 3 Here 1 1 cosπx = exp S π 3 S 1 = exp π = 3 3 = 1 3 ζ3 Hece we obtai 3 Let We prove that S 3 x = e x = 3 ζ3 1 x e x S 3 x = C S 3 x S x 3 S 1 x with C = e ζ The we get usig the product expressios for S 3 x, S x ad S 1 x First, we see S 3 S 3 x = πx cotπx 15

16 from [1, 13, 1] ad S 3 x 3 S x + S 1 x S 3 S S 1 = x 1x π + 3x cotπx = πx cotπx Hece, it is sufficiet to show that C = S 3 1 S 3 1 We calculated the value S 3 1 i [16] [13] as S 3 1 S S 1 = 1 exp 7 8π ζ3, which is equivalet to the followig result of Euler [3]: We kow ζ3 = π 7 log S 3 1 π 0 x logsi xdx = 38 exp 3 16π ζ3 as above Thus we have S 3 S 3 S S 1 = 1 exp 7 8π ζ3 38 exp 3 16π ζ3 3 1 = exp ζ3 π = expζ = C, where we used ζ3 = π ζ comig from the fuctioal equatio for ζs Remark For aother approach to Theorems 6 ad 7 we refer to [13], where we use multiplicatio formulas for multiple sie fuctios ad the argumet is more elaborate The above proofs are quite direct because of our Kummer s formula for multiple gamma fuctios 16

17 5 Geeralizatios ad Problems Our ivestigatio aturally leads to the case of Γ r x for r This is treated similarly as r = ad 3 by usig r 1 1 ζ r s, x = r 1! ζs r + 1, x + b r,k xζ k s, x where b r,k x is a polyomial i x determied by + r 1 = r 1 for all itegers 0 For example + xr 1 r 1! k=1 r 1 + k 1 + b r,k x k 1 k=1 The we have b r,1 x = x 1r 1 r 1! Γ r x = exp ζ r0, x 1 = exp r 1! ζ 1 r, x r 1 Γ k x br,kx Hece our Theorem 3 gives the Kummer s formula for Γ r x iductively Beyod this we have a more geeral problem for Γ r x; ω 1,, ω r with geeral periods ω 1,, ω r defied by where k=1 Γ r x; ω 1,, ω r = exp ζ r0, x, ω 1,, ω r, ζ r s, x, ω 1,, ω r = 1,, r =0 1 ω r ω r + x s is the multiple Hurwitz zeta fuctio for geeral parameters We have see the case ω 1,, ω r = 1,, 1 above First, whe ω 1,, ω r is reduced to the ratioal parameters the situatio is quite similar to the case 1,, 1 For example, if ω 1, ω = 1,, the ζ s, x, 1, = s ζ s, x + ζ s, x + 1 ad ζ 0, x, 1, = ζ 0, x + ζ 0, x + 1 log ζ 0, x + ζ 0, x

18 see [15] Sice ζ 0, x + ζ 0, x + 1 = ζ, 0, x, 1, = 1 x 3x + 11, 6 we have log Γ x, 1, = log Γ x x log Γ log x 3x Hece we obtai Kummer s formula for Γ x, 1, from our Theorem 11 The o-ratioal or o-commesurable parameter case, we face the problem to have the fuctioal equatio for ζ r s, x, ω 1,, ω r of the form ζ r s, x, ω 1,, ω r = ξ r r s, x, ω 1,, ω r by usig the residue calculatio as i Riema [] ad Hurwitz [8] This problem is i geeral highly o-trivial delicate covergece as ivestigated by Hardy [5, 6] After that we would have the desired Kummer s type formula: log Γ r x; ω 1,, ω r = ξ rr, x, ω 1,, ω r We postpoe the detailed ivestigatio to the ext opportuity Refereces [1] EW Bares: O the theory of the multiple gamma fuctio Tras Cambridge Philos Soc, [] S Chowla ad A Selberg O Epstei s zeta-fuctio J reie agew Math Crelle s J [3] L Euler: Exercitatioes aalyticae Novi Commetarii Academiae Scietiarum Petropolitaae, Opera Omia I-15, pp [] GH Hardy O Kummer s series for log Γa Quarterly J Math Collected Papers: Vol IV, pp 8-3 [5] GH Hardy The expressio of the double zeta-fuctio ad double gamma-fuctio i terms of elliptic fuctios Tras Cambridge Phil Soc Collected Papers: Vol IV, pp

19 [6] GH Hardy O double Fourier series, ad especially those which represet the double zeta-fuctio with real ad icommesurable parameters Quarterly J Math Collected Papers: Vol IV, pp [7] O Hölder: Ueber eie trascedete Fuctio Göttige Nachrichte 1886, Nr 16 pp 51-5 [8] A Hurwitz Eiige Eigeschafte der Dirichlet sche Fuktioe F s = D 1 s, die bei der Bestimmug der Klasseazahle biärer quadratischer Forme auftrete Zeitschrift für Mathematik ud Physik Werke Vol I, pp 7-88 [9] E Kummer Beitrag zur Theorie der Fuctio Γx = 0 e v v x 1 dv J reie agew Math Crelle s J Collected Papers Vol II, pp35-38 [10] N Kurokawa: Multiple sie fuctios ad Selberg zeta fuctios Proc Japa Acad 67A [11] N Kurokawa: Gamma factors ad Placherel measures Proc Japa Acad 68A [1] N Kurokawa: Multiple zeta fuctios: a example I Zeta Fuctios i Geometry, volume 1 of Advaced Studies i Pure Math, pages 19-6, Kiokuiya, Tokyo 199 [13] N Kurokawa ad S Koyama: Multiple sie fuctios Forum Math i press [1] S Koyama ad N Kurokawa: Multiple zeta fuctios I Composit Math i press [15] S Koyama ad N Kurokawa: Normalized double sie fuctios Proc Japa Acad [16] N Kurokawa ad M Wakayama: O ζ3 J Ramauja Math Soc [17] E Ladau Über die zu eiem algebraische Zahlkörper gehörige Zetafuctio ud die Ausdehug der Tschebyschefsche Primzahltheorie auf das Problem der Verteilug der Primideale J reie agew Math Crelle s J Collected Works Vol I, pp [18] M Lerch Dalši studie v oboru Malmstéovských řad Rozpravy České Akad o 8, pp 1-61 [19] M Lerch Sur quelques formules relatives du ombre des classes Bull Sci Math

20 [0] CJ Malmsté De itegralibus quibusdam defiitis, seriebusque ifiitis J reie agew Math Crelle s J [1] Yu I Mai: Lectures o zeta fuctios ad motives accordig to Deiger ad Kurokawa Asterisque [] B Riema: Ueber die Azahl der Primzahle uter eier gegebee Grösse Moatsberichte der Berlier Akademie, November 1859, pp Gesammelte Mathematische Werke, pp [3] P Sarak: Determiats of Laplacias Comm Math Phys [] T Shitai: O a Kroecker limit formula for real quadratic fields J Fac Sci Uiv Tokyo, [5] A Voros: Spectral fuctios, special fuctios ad Selberg trace formula Comm Math Phys Shi-ya Koyama: Departmet of Mathemartics, Keio Uiversity, 3-1-1, Hiyoshi, Kohokuku, Kaagawa 3-85, Japa koyama@mathkeioacjp Nobushige Kurokawa: Departmet of Mathematics, Tokyo Istitute of Techology, -1-1, Oh-okayama, Meguro-ku, Tokyo , Japa kurokawa@mathtitechacjp 0