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RAMANUJAN SERIES UPSIDE-DOWN

Part of: Discontinuous groups and automorphic forms Computational number theory Other special functions

Published online by Cambridge University Press:  07 July 2014

JESUS GUILLERA  and
MATHEW ROGERS
JESUS GUILLERA
Affiliation:
Av. Cesáreo Alierta, 31 esc. izda 4°–A, Zaragoza, Spain email [email protected]
MATHEW ROGERS*
Affiliation:
Department of Mathematics and Statistics, Université de Montréal, CP 6128 succ. Centre-ville, Montréal Québec H3C 3J7,Canada email [email protected]
*
✉[email protected]
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Abstract

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We prove that there is a correspondence between Ramanujan-type formulas for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet $L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

Keywords

Dirichlet L-valuesformulas for πhypergeometric serieslattice sumsRamanujan

MSC classification

Secondary: 33C20: Generalized hypergeometric series, $_pF_q$ 11F11: Holomorphic modular forms of integral weight 11F03: Modular and automorphic functions 11Y60: Evaluation of constants 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 1 , August 2014 , pp. 78 - 106
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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