Asymptotic second moment of Dirichlet $L$-functions along a thin coset
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913089601929216 |
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| author | Garcia, Bradford Young, Matthew P. |
| author_facet | Garcia, Bradford Young, Matthew P. |
| contents | We prove an asymptotic formula for the second moment of central values of Dirichlet $L$-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo $d$ inside the full group of characters modulo $q$. Suppose that $ν_p(d) \geq ν_p(q)/2$ for all primes $p$ dividing $q$. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size $q^{1/2}$ here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem.
We also obtain an asymptotic result for smaller $d$, with $ν_p(q)/3 \leq ν_p(d) \leq ν_p(q)/2$, with a power-saving error term for $d$ larger than $q^{2/5}$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughly $d$, which is only slightly smaller than the diagonal main term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08482 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic second moment of Dirichlet $L$-functions along a thin coset Garcia, Bradford Young, Matthew P. Number Theory 11M06 We prove an asymptotic formula for the second moment of central values of Dirichlet $L$-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo $d$ inside the full group of characters modulo $q$. Suppose that $ν_p(d) \geq ν_p(q)/2$ for all primes $p$ dividing $q$. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size $q^{1/2}$ here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smaller $d$, with $ν_p(q)/3 \leq ν_p(d) \leq ν_p(q)/2$, with a power-saving error term for $d$ larger than $q^{2/5}$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughly $d$, which is only slightly smaller than the diagonal main term. |
| title | Asymptotic second moment of Dirichlet $L$-functions along a thin coset |
| topic | Number Theory 11M06 |
| url | https://arxiv.org/abs/2312.08482 |