Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912231597277184 |
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| author | Kaneko, Ikuya |
| author_facet | Kaneko, Ikuya |
| contents | A new reciprocity formula for Dirichlet $L$-functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$-functions in the $t$-aspect to the cubic moment of central $L$-values of Hecke-Maass newforms of level at most $q^{2}$ and primitive central character $ψ^{2}$ averaged over all primitive nonquadratic characters $ψ$ modulo $q$. Our formula can be thought of as a reverse version of recent work of Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval fourth moment bound and the twelfth moment bound for Dirichlet $L$-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_08974 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications Kaneko, Ikuya Number Theory Representation Theory 11M06 (primary), 11F03, 11F72 (secondary) A new reciprocity formula for Dirichlet $L$-functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$-functions in the $t$-aspect to the cubic moment of central $L$-values of Hecke-Maass newforms of level at most $q^{2}$ and primitive central character $ψ^{2}$ averaged over all primitive nonquadratic characters $ψ$ modulo $q$. Our formula can be thought of as a reverse version of recent work of Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval fourth moment bound and the twelfth moment bound for Dirichlet $L$-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms. |
| title | Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications |
| topic | Number Theory Representation Theory 11M06 (primary), 11F03, 11F72 (secondary) |
| url | https://arxiv.org/abs/2110.08974 |