A note on special values of 𝐿-functions

Gun, Sanoli; Murty, M.; Rath, Purusottam

doi:10.1090/S0002-9939-2014-11858-2

A note on special values of $L$-functions
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by Sanoli Gun, M. Ram Murty and Purusottam Rath

Proc. Amer. Math. Soc. 142 (2014), 1147-1156

DOI: https://doi.org/10.1090/S0002-9939-2014-11858-2

Published electronically: January 30, 2014

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Abstract:

In this paper, we link the nature of special values of certain Dirichlet $L$-functions to those of multiple gamma values.

References

V. S. Adamchik, The multiple gamma function and its application to computation of series, Ramanujan J. 9 (2005), no. 3, 271–288. MR 2173489, DOI 10.1007/s11139-005-1868-3
Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004 (French, with English and French summaries). MR 2115000
E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374–425.
Cristiana Bertolin, Périodes de 1-motifs et transcendance, J. Number Theory 97 (2002), no. 2, 204–221 (French, with English summary). MR 1942957, DOI 10.1016/S0022-314X(02)00002-1
Daniel Bertrand and David Masser, Linear forms in elliptic integrals, Invent. Math. 58 (1980), no. 3, 283–288. MR 571576, DOI 10.1007/BF01390255
Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, Mathematical Surveys and Monographs, vol. 19, American Mathematical Society, Providence, RI, 1984. MR 772027, DOI 10.1090/surv/019
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
J. Dufresnoy and Ch. Pisot, Sur la relation fonctionnelle $f(x+1)-f(x)=\varphi (x)$, Bull. Soc. Math. Belg. 15 (1963), 259–270 (French). MR 161055
William Duke and Özlem Imamoḡlu, Special values of multiple gamma functions, J. Théor. Nombres Bordeaux 18 (2006), no. 1, 113–123 (English, with English and French summaries). MR 2245878
J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta (1/2+it)$, Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265, DOI 10.1007/s002200000261
Serge Lang, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0214547
Brad A. Lutes and Matthew A. Papanikolas, Algebraic independence of values of Goss $L$-functions at $s=1$, J. Number Theory 133 (2013), no. 3, 1000–1011. MR 2997783, DOI 10.1016/j.jnt.2012.04.002
David Masser, Elliptic functions and transcendence, Lecture Notes in Mathematics, Vol. 437, Springer-Verlag, Berlin-New York, 1975. MR 0379391
Yu. V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65–96 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 9, 1319–1348. MR 1422383, DOI 10.1070/SM1996v187n09ABEH000158
Introduction to algebraic independence theory, Lecture Notes in Mathematics, vol. 1752, Springer-Verlag, Berlin, 2001. With contributions from F. Amoroso, D. Bertrand, W. D. Brownawell, G. Diaz, M. Laurent, Yuri V. Nesterenko, K. Nishioka, Patrice Philippon, G. Rémond, D. Roy and M. Waldschmidt; Edited by Nesterenko and Philippon. MR 1837822, DOI 10.1007/b76882
P. Philippon, Variétés abéliennes et indépendance algébrique. II. Un analogue abélien du théorème de Lindemann-Weierstraß, Invent. Math. 72 (1983), no. 3, 389–405 (French). MR 704398, DOI 10.1007/BF01398395
T. Rivoal and W. Zudilin, Diophantine properties of numbers related to Catalan’s constant, Math. Ann. 326 (2003), no. 4, 705–721. MR 2003449, DOI 10.1007/s00208-003-0420-2
Peter Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120. MR 885573
Takuro Shintani, A proof of the classical Kronecker limit formula, Tokyo J. Math. 3 (1980), no. 2, 191–199. MR 605088, DOI 10.3836/tjm/1270472992
Ilan Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19 (1988), no. 2, 493–507. MR 930041, DOI 10.1137/0519035
Marie-France Vignéras, L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire $\textrm {PSL}(2,\,\textbf {Z})$, Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978) Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 235–249 (French). MR 556676
G. Wüstholz, Über das Abelsche Analogon des Lindemannschen Satzes. I, Invent. Math. 72 (1983), no. 3, 363–388 (German). MR 704397, DOI 10.1007/BF01398393