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# On M-functions associated with modular forms

Abstract : Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or $(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $\chi$ or $f$ vary. First, for a quasi-character $\psi\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit for the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi, s)),$ when $f$ is fixed and $\chi$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes \chi,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h_f \psi(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimes\chi, s).$
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https://hal.archives-ouvertes.fr/hal-01475616
Contributor : Alexey Zykin Connect in order to contact the contributor
Submitted on : Thursday, February 23, 2017 - 9:22:46 PM
Last modification on : Wednesday, August 24, 2022 - 11:35:17 AM
Long-term archiving on: : Wednesday, May 24, 2017 - 2:26:10 PM

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### Citation

Philippe Lebacque, Alexey Zykin. On M-functions associated with modular forms. Moscow Mathematical Journal, 2018, 18 (3), pp.437-472. ⟨10.17323/1609-4514-2018-18-3-437-472⟩. ⟨hal-01475616⟩

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