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One-Level density for holomorphic cusp forms of arbitrary level

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arxiv 1604.03224 v4 pith:AX3AE3O2 submitted 2016-04-12 math.NT math-phmath.MP

classification math.NTmath-phmath.MP
keywords densitylevelone-levelarbitrarycertainholomorphicsquare-freeappropriate
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abstract

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Mili\'cevi\'c, which is of use for other problems as well.

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  1. Squarefree numbers in short intervals: explicit and formalized

    math.NT 2026-08 conditional novelty 6.0 of 10

    For intervals of length H = X^{1/5 - 2/90935 + ε}, the number of squarefree integers differs from (6/π²)H by at most an explicit constant times H X^{-ε/10^{25}}.

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