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A NOTE ON SPECIAL VALUES OF CERTAIN DIRICHLET L-FUNCTIONS

  • B. RAMAKRISHNAN (a1)
Abstract

In Gun and Ramakrishnan [‘On special values of certain Dirichlet L-functions’, Ramanujan J.15 (2008), 275–280], we gave expressions for the special values of certain Dirichlet L-function in terms of finite sums involving Jacobi symbols. In this note we extend our earlier results by giving similar expressions for two more special values of Dirichlet L-functions, namely L(−1,χm) and L(−2,χm′), where m,m′ are square-free integers with m≡1 mod 8 and m′≡3 mod 8 and χD is the Kronecker symbol . As a consequence, using the identities of Cohen [‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann.217 (1975), 271–285], we also express the finite sums with Jacobi symbols in terms of sums involving divisor functions. Finally, we observe that the proof of Theorem 1.2 in Gun and Ramakrishnan (as above) is a direct consequence of Equation (24) in Gun, Manickam and Ramakrishnan [‘A canonical subspace of modular forms of half-integral weight’, Math. Ann.347 (2010), 899–916].

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References
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[1]Bateman, P. T. and Knopp, M. I., ‘Some new old-fashioned modular identities’, Ramanujan J. 2 (1998), 247269.
[2]Cohen, H., ‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann. 217 (1975), 271285.
[3]Cooper, S., ‘Sums of five, seven and nine squares’, Ramanujan J. 6 (2002), 469490.
[4]Cooper, S. and Hirschhorn, M., ‘On the number of primitive representations of integers as sums of squares’, Ramanujan J. 13 (2007), 725.
[5]Gun, S., Manickam, M. and Ramakrishnan, B., ‘A canonical subspace of modular forms of half-integral weight’, Math. Ann. 347 (2010), 899916.
[6]Gun, S. and Ramakrishnan, B., ‘On special values of certain Dirichlet L-functions’, Ramanujan J. 15 (2008), 275280.
[7]Gun, S. and Ramakrishnan, B., ‘On the representation of integers as sums of an odd number of squares’, Ramanujan J. 15 (2008), 367376.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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