arXiv · 2203.05296
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
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Daniel Duverney, Iekata Shiokawa. 2022-03-10. A new proof of a theorem of Tietze. https://arxiv.org/abs/2203.05296
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We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.
We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box that satisfy the determinant equation $x_1x_2-x_3x_4 =r$ for $r \neq 0 $ with two of entries to be prime. Finally, these estimates are applied to evaluate mixed fourth moments of Dirichlet character sums over integers and primes, yielding non-trivial bounds. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes.
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