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Abstract We prove an asymptotic formula for the second moment of central values of DirichletL-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulodinside the full group of characters moduloq. Suppose that$$\nu _p(d) \geq \nu _p(q)/2$$for all primespdividingq. 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 smallerd, with$$\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$$, with a power-saving error term fordlarger than$$q^{2/5}$$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughlyd, which is only slightly smaller than the diagonal main term.
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On the error term in a mixed moment of L ‐functions
Abstract There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of DirichletL‐functions and a closely related mixed moment ofL‐functions involving automorphicL‐functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.
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- PAR ID:
- 10430981
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Mathematika
- Volume:
- 69
- Issue:
- 3
- ISSN:
- 0025-5793
- Page Range / eLocation ID:
- p. 573-583
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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