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Abstract We compute moments of L-functions associated to the polynomial family of Artin–Schreier covers over $$\mathbb{F}_q$$, where q is a power of a prime p > 2, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $$k{\text{th}}$$ moment for a large range of values of k, depending on the sizes of p and q. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family.
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Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in {\zeta(s)}. For example, integrating {|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto {2T}, we obtain an asymptotic formula when the shift α is roughly bigger than {\frac{1}{\log T}}and {k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as {\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function.more » « less
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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 $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$.more » « less
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Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.more » « less
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