Breaking the12-barrier for the twisted second moment of Dirichlet L-functions

Abstract

We study the second moment of Dirichlet L-functions to a large prime modulus q twisted by the square of an arbitrary Dirichlet polynomial. We break the 1 2 -barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q 51/101 = q 1/2+1/202 . As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet L-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

Bounds for moments of Dirichlet $L$-functions to a fixed modulus

“…Moreover, the result is shown to be valid for k = 3/2 as well by H. M. Bui, K. Pratt, N. Robles and A. Zaharescu [3,Theorem 1.4]. This case is achieved by employing various tools including a result on a long mollified second moment of the corresponding family of L-functions given in [3,Theorem 1.1]. In our proofs of Theorems 1.1 and 1.2, we also need to evaluate certain twisted second moments for the same family.…”

“…In particular, the case k = 1 of (1.4) is explicitly given in [4]. Moreover, the result is shown to be valid for k = 3/2 as well by H. M. Bui, K. Pratt, N. Robles and A. Zaharescu [3,Theorem 1.4]. This case is achieved by employing various tools including a result on a long mollified second moment of the corresponding family of L-functions given in [3,Theorem 1.1].…”

“…Hence, only the orthogonality relation for characters is needed to complete our work. We also point out here that as it is mentioned in [3] that one may apply the work of B. Hough [12] or R. Zacharias [24] on twisted fourth moment for the family of Dirichlet L-functions modulo q to obtain sharp upper bounds on all moments below the fourth. We decide to use the twisted second moment here to keep our exposition simple by observing that it is needed for obtaining both the lower bounds and the upper bounds.…”

Weighted central limit theorems for central values of $L$-functions

“…The estimate for the odd characters follows from a similar argument. We quote the following twisted second moment of Dirichlet L-functions (see [25] and [8,Theorem 1.1]). Lemma 7.1.…”

Exceptional characters and nonvanishing of Dirichlet $L$-functions

“…In contrast to recent works on the mollified moments of L(1/2, χ) (e.g. [27,15,2,28,5]), however, we do not appeal to the spectral theory of automorphic forms. The reason for this is that, instead of utilizing an approximate function equation for |L χ (1/2)| 2 , we obtain an expression for |L χ (1/2)| 2 by squaring the approximate functional equation for L χ (1/2).…”