Abstract
Motivated by the classical work of Ramanujan and recent work of Berndt and Zaharescu, we establish certain infinite families of identities relating values of Dirichlet L-functions at positive integers to Gauss sums and trigonometric sums twisted with characters.
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Alkan, E.: On Dirichlet L-functions with periodic coefficients and Eisenstein series. Monatsh. Math. (2011, to appear)
Beck, M., Berndt, B.C., Chan, O.-Y., Zaharescu, A.: Determinations of analogues of Gauss sums and other trigonometric sums. Int. J. Number Theory 1, 333–356 (2005)
Berndt, B.C.: Character analogues of the Poisson and Euler–MacLaurin summation formulas with applications. J. Number Theory 7, 413–445 (1975)
Berndt, B.C.: Ramanujan’s Notebooks Part IV. Springer, New York (1994)
Berndt, B.C., Schoenfeld, L.: Periodic analogues of the Euler–MacLaurin and Poisson summation formulas with applications to number theory. Acta Arith. 28, 23–68 (1975/76)
Berndt, B.C., Zaharescu, A.: Finite trigonometric sums and class numbers. Math. Ann. 330, 551–575 (2004)
Berndt, B.C., Zhang, L.-C.: Ramanujan’s identities for eta-functions. Math. Ann. 292, 561–573 (1992)
Berndt, B.C., Zhang, L.-C.: A new class of theta-function identities originating in Ramanujan’s notebooks. J. Number Theory 48, 224–242 (1994)
Chan, O.-Y.: Weighted trigonometric sums over a half period. Adv. Appl. Math. 38, 482–504 (2007)
Davenport, H.: Multiplicative Number Theory, 3rd edn. Graduate Texts in Mathematics, vol. 74. Springer, New York (2000)
Dirichlet, G.L.: Recherches sur diverses applications de l’analyse infinitésimale à le théorie des nombres, seconde partie. J. Reine Angew. Math. 21, 134–155 (1840)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)
Lebesgue, V.A.: Suite du Memorie sur les applications du symbole \((\frac {a}{b})\). J. Math. 15, 215–237 (1850)
Liu, Z.-G.: Some Eisenstein series identities related to modular equations of the seventh order. Pac. J. Math. 209, 103–130 (2003)
Murty, M.R.: Problems in Analytic Number Theory. Graduate Texts in Mathematics, vol. 206. Springer, New York (2001)
Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften, vol. 322. Springer, New York (1999)
Ramanujan, S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957)
Schemmel, V.: De Multitudine formarum secundi gradus disquisitiones. Dissertation, Breslau (1863)
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The author is supported by the Distinguished Young Scholar Award, Tüba-Gebip of Turkish Academy of Sciences.
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Alkan, E. Values of Dirichlet L-functions, Gauss sums and trigonometric sums. Ramanujan J 26, 375–398 (2011). https://doi.org/10.1007/s11139-010-9292-8
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DOI: https://doi.org/10.1007/s11139-010-9292-8