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Mean value of cubic $L$-funcitons with fixed genus

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arxiv 2503.17228 v3 pith:ILTCHWVR submitted 2025-03-21 math.NT

Ziwei Hong , Zhongqiu Fang This is my paper · ORCID
classification math.NT
keywords cubicprimitiveequationfracgenusmeanvalueassociated
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abstract

We investigate the mean value of the first moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ cubic\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi), \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive cubic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{7}{8}+\varepsilon)g}$.

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  1. Twisted second moment of primitive cubic L-functions

    math.NT 2025-06 reject novelty 6.0 of 10

    A claimed twisted second moment asymptotic for primitive cubic L-functions over F_q(T) is invalidated by a sign error in the residue-theoretic main term.

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