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## nt.quantity idea – Distribution of zeros of actual quadratic Dirichlet L-functions in diminutive intervals

**Motivation:** Some information gathered on least quadratic nonresidues explain that the zeros of quadratic Dirichlet L-functions are extra evenly spaced than that in common Dirichlet L-functions.

Question.Let $chi$ breathe a actual quadratic Dirichlet personality. Assuming the Generalized Riemann Hypothesis, what number of zeros of the

quadratic L-function $L(s,chi)$ does the area ${1/2+textual content{i} t:

a<t<b}$ acquire?

The respond closely is determined by the relative dimension of $a$, $b$ and $|a-b|$ with respect to the conductor of $chi$, $d(chi)$.

In this query, we occupy $|a-b|=Theta(1/log log d(chi))$, and $a$, $b$ take values in $[0,f(chi)]$, the place $f$ is a few operate that grows to infinity as $d(chi) rightarrow infty$. There are not any restrictions on the expansion fee of $f$.

A simple density appraise offers the density $approx 1/2pi log (t/2pi) + log d(chi)$ at $L(1/2+textual content{i}t,chi)$. Thus an affordable respond ought to restrict its errors on the variety of zeros inside $o(log d(chi)/loglog d(chi))$.

I might anticipate a power-law cancellation of zeros, i.e. the mistake time period is $o(log^{alpha} d(chi))$ for some absolute $alpha<1$.

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