The result is conditional
A mathematical preprint reports an explicit asymptotic formula for a mixed second moment, a way of summarizing the combined size of many L-function values, at the central point. The result carries a power-saving error term, meaning the remainder shrinks by a fixed power as the relevant parameter grows. The conclusion is conditional: it assumes Selberg’s eigenvalue conjecture and applies only to the paper’s specified family of moduli and characters.
The family is deliberately structured. It averages all primitive even Dirichlet characters modulo q, while q must be split as q=q1q2 inside the paper’s set Q. The two factors have the stated upper bounds Q1 and Q2, with Q2 comparable to Q raised to δ1; the setup also requires (q,6)=1, (q1,q2)=1 and 0<δ1<0.0004.
What the theorem says
Under Selberg’s eigenvalue conjecture, the unweighted mixed second moment is asymptotic to an explicit arithmetic main term. The expression combines an arithmetic constant with special values of the fixed form and its symmetric-square L-function, a factor depending on the modulus, the count of primitive even characters, and a squared logarithm. The fixed form is a Hecke holomorphic cusp form for the full modular group, with weight κ and normalized Hecke eigenvalues.
The paper also states a weighted version. It formulates the asymptotic for smooth functions Ψ1 and Ψ2 supported on the interval [1,2], under the stated conditions on the modulus factors. In other words, the theorem is written for both the unweighted family and a version in which the two modulus factors are selected through smooth weights.
How the proof is built
To obtain the result, the proof uses the approximate functional equation for L-functions to turn evaluation of the mixed moment into an estimation problem. Its off-diagonal treatment uses divisor switching, and the paper explicitly divides the analysis into two main cases. The authors then combine these steps with a generalized p-adic stationary-phase method and earlier ideas in complementary regimes to cover the unbalanced range.
The authors identify the generalized p-adic stationary-phase method as the main innovation. They say it works with earlier ideas in complementary regimes to cover the unbalanced range, a central part of the proof’s treatment of the factored moduli.
Several auxiliary estimates support the central asymptotic. One stated theorem gives an upper bound for prime-power hyper-Kloosterman sums under its listed prime and exponent conditions. A further estimate roughly yields square-root cancellation under stated coprimality conditions. The paper also establishes a new large-sieve inequality used in the proof. These estimates are supporting results rather than separate mixed-moment asymptotics.
The boundaries of the claim
The restrictions matter. The main result assumes Selberg’s eigenvalue conjecture and a suitable factorization q=q1q2, including the stated size and coprimality conditions. It is formulated for a fixed Hecke holomorphic cusp form and primitive even characters. The authors identify removing the factorization assumption as future research.
The result remains conditional rather than unconditional, and the supplied analysis does not extend it to arbitrary moduli without the factorization restriction. The supplied document is an arXiv preprint, and no funding information is reported in the supplied paper text.
Paper data and sources
Original title: The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point
Authors: Zhengye Chen, Yongxiao Lin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text