Abstract
We study the $1$-level density and the pair correlation of zeros of quadratic Dirichlet $L$-functions in function fields, as we average over the ensemble $\mathcal{H}_{2g+1}$ of monic, square-free polynomials with coefficients in $\mathbb{F}_q[x]$. In the case of the $1$-level density, when the Fourier transform of the test function is supported in the restricted interval $(\frac{1}{3},1)$, we compute a secondary term of size $q^{-\frac{4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in $(\frac{1}{3}, \frac{1}{2})$, we identify another lower order term of size $q^{-\frac{8g}{5}}/g$. We also compute the pair correlation, and as for the $1$-level density, we detect lower order terms under certain restrictions; for example, we see a term of size $q^{-g}/g^2$ when the Fourier transform is supported in $(\frac{1}{4},\frac{1}{2})$. The $1$-level density and the pair correlation allow us to obtain non-vanishing results for $L(\frac12,\chi_D)$, as well as lower bounds for the proportion of simple zeros of this family of $L$-functions.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.