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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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Two problems on the distribution of Carmichael's lambda function
Abstract Let denote the exponent of the multiplicative group modulon. We show that whenqis odd, each coprime residue class moduloqis hit equally often by asnvaries. Under the stronger assumption that , we prove that equidistribution persists throughout a Siegel–Walfisz‐type range of uniformity. By similar methods we show that obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order ofamodn, for any fixed integer .
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- Award ID(s):
- 2001581
- PAR ID:
- 10465630
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Mathematika
- Volume:
- 69
- Issue:
- 4
- ISSN:
- 0025-5793
- Format(s):
- Medium: X Size: p. 1195-1220
- Size(s):
- p. 1195-1220
- Sponsoring Org:
- National Science Foundation
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