The first moment of quadratic twists of modular $L$-functions

Abstract

We obtain the asymptotic formula with an error term O(X 1 2 +ε ) for the smoothed first moment of quadratic twists of modular L-functions. We also give a similar result for the smoothed first moment of the first derivative of quadratic twists of modular L-functions. The argument is largely based on Young's recursive method [19,20].

Lower bounds for moments of quadratic twists of modular $L$-functions

“…It follows from [17, Conjecture 3, Theorem 1.4] and the proofs of Theorems 1.1 and 1.2 in [17] that for…”

“…If κ ≡ 0 (mod 4), we set α = 0 in (3.6) and compute Z( 1 2 , l) using (3.7) to deduce readily (3.4). When κ ≡ 2 (mod 4), we differentiate both sides of (3.6) with respect to α by noticing that the contribution of the derivative of the error term is still O(l 1/2+ε X 1/2+ε ) using Cauchy's integral formula and the observation that the error term in (3.6) is holomorphic on the disc centred at (0, 0) with radius ≪ (log X) −1 from the proof of [17,Theorem 1.4]. Upon setting α = 0, we obtain that *…”

“…In [21,22], M. P. Young developed a recursive method to improve the error terms in the first and third moment of quadratic Dirichlet L-functions. This method was adapted by Q. Shen [17] to ameliorate the error terms in the first moment of quadratic twists of modular L-functions. The results of Shen can be regarded as commensurate to a result of K. Sono [18] on the second moment of quadratic Dirichlet L-functions.…”

“…Our proof of the above theorem is mainly based on the lower bounds principle of Heap and Soundararajan in [5]. As this method requires the evaluation of the twisted moments of modular functions, we shall also make crucial use of the results of Q. Shen [17] on the twisted first moment of twisted modular functions.…”