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

TL;DR

This paper derives an asymptotic formula with a precise error term for the average behavior of quadratic twists of modular L-functions and their derivatives, advancing understanding in analytic number theory.

Contribution

It provides the first asymptotic formulas with explicit error bounds for the first moments of quadratic twists and their derivatives of modular L-functions.

Findings

Asymptotic formula with error term O(X^{1/2 + ε}) for the first moment

Similar result obtained for the first derivative of quadratic twists

Method based on Young's previous works

Abstract

We obtain an asymptotic formula with an error term $O(X^{\frac{1}{2} + \varepsilon})$ 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 works (Acta Arith 138(1):73-99, 2009 and Selecta Math 19(2):509-543, 2013).