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Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$.
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Character Ratios for Exceptional Groups of Lie Type
Abstract We prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $$\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$$ for all nontrivial irreducible characters $$\chi$$ and nonidentity elements $$g$$, where $$c$$ is an absolute constant, and $$k$$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs.
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- Award ID(s):
- 1840702
- PAR ID:
- 10121232
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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