Lower bounds for negative moments of quadratic Dirichlet $L$-functions

Abstract

We establish lower bounds for the 2k-th moment of families of quadratic Dirichlet L-functions at the central point for all real k < 0, assuming a conjecture of S. Chowla on the non-vanishing of these L-values.

Negative first moment of quadratic twists of $L$-functions

“…In [7], asymptotic formulas are achieved for these negative moments for certain ranges of the shifts over function fields. Additionally, lower bounds are given in [9] for negative moments of families of quadratic Dirichlet L-functions at the central point in the number field setting, assuming the truth of a conjecture of S. Chowla [6] on the non-vanishing of these L-values. Note that the results in [9] are only expected to be sharp for the 2k-th moment with −5/2 ≤ k < 0 since the work in [8] also suggests certain phase changes in the asymptotic formulas for the 2k-th moment of the family of quadratic Dirichlet L-functions when 2k = −(2j + 1/2) for any positive integer j.…”

“…Additionally, lower bounds are given in [9] for negative moments of families of quadratic Dirichlet L-functions at the central point in the number field setting, assuming the truth of a conjecture of S. Chowla [6] on the non-vanishing of these L-values. Note that the results in [9] are only expected to be sharp for the 2k-th moment with −5/2 ≤ k < 0 since the work in [8] also suggests certain phase changes in the asymptotic formulas for the 2k-th moment of the family of quadratic Dirichlet L-functions when 2k = −(2j + 1/2) for any positive integer j. We also point out here that asymptotic formulas for negative moments of the above family of L-functions at 1 were obtained by A. Granville and K. Soundararajan [11] in the number field setting and by A. Lumley [14] for function fields.…”

Sharp lower bounds for negative moments of quadratic twist of modular L-functions

“…According to a conjecture by S. Chowla [16] in his work on the quadratic Dirichlet L-functions, it is suggested that L( 1 2 , χ) should not equal to zero for all primitive real Dirichlet characters χ. Assuming S. Chowla's conjecture, Peng Gao [5] establishes lower bounds for the 2k-th moment of quadratic Dirichlet L-functions for all real k < 0. The estimation is given by…”