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We prove an asymptotic formula for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q, over all moduli q≤Q and with a short average on the critical line. Previously the same result was shown conditionally on the Generalized Riemann Hypothesis by the first two authors.
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This content will become publicly available on February 1, 2026
A Mean Value Theorem for Dirichlet Polynomials Associated With Primitive Dirichlet L -Functions
Consider the family of Dirichlet $$L$$-functions of all even primitive characters of conductor at most $$Q$$, where $$Q$$ is a parameter tending to infinity. For $$X=Q^{\eta }$$ with $$1<\eta <2$$, we examine Dirichlet polynomials of length $$X$$ with coefficients those of the Dirichlet series of a product of an arbitrary (finite) number of shifted $$L$$-functions from the family. Assuming the Generalized Lindelöf Hypothesis for Dirichlet $$L$$-functions, we prove an asymptotic formula for averages of these Dirichlet polynomials. Our result agrees with the prediction of the recipe of Conrey, Farmer, Keating, Rubinstein, and Snaith for these averages. One may view our result as evidence for the “one-swap” terms in the recipe prediction for the general $2k$th moment of the family of Dirichlet $$L$$-functions.
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- PAR ID:
- 10629109
- Publisher / Repository:
- International Mathematics Research Notices
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 3
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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