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nt.quantity concept – Zero density estimates of actual quadratic Dirichlet L-functions in diminutive intervals Answer

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nt.quantity concept – Zero density estimates 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 relies on 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 perform that grows to infinity as $d(chi) rightarrow infty$. There aren’t any restrictions on the expansion fee of $f$.

A simple density appraise provides the density $approx 1/2pi log (t/2pi) + log d(chi)$ at $L(1/2+textual content{i}t,chi)$. Thus an inexpensive 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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