Distribution of zeros of real quadratic Dirichlet L-functions in small intervals

Question

Motivation: Some data gathered on least quadratic nonresidues indicate that the zeros of quadratic Dirichlet L-functions are more evenly spaced than that in general Dirichlet L-functions.

Question. Let $\chi$ be a real quadratic Dirichlet character. Assuming the Generalized Riemann Hypothesis, how many zeros of the quadratic L-function $L(s,\chi)$ does the region $\{1/2+\text{i} t: a<t<b\}$ contain?

The answer heavily depends on the relative size of $a$, $b$ and $|a-b|$ with respect to the conductor of $\chi$, $d(\chi)$.

In this question, we assume $|a-b|=\Theta(1/\log \log d(\chi))$, and $a$, $b$ take values in $[0,f(\chi)]$, where $f$ is some function that grows to infinity as $d(\chi) \rightarrow \infty$. There are no restrictions on the growth rate of $f$.

A naive density estimate gives the density $\approx 1/2\pi \log (t/2\pi) + \log d(\chi)$ at $L(1/2+\text{i}t,\chi)$. Thus a reasonable answer should limit its errors on the number of zeros within $o(\log d(\chi)/\log\log d(\chi))$.

I would expect a power-law cancellation of zeros, i.e. the error term is $o(\log^{\alpha} d(\chi))$ for some absolute $\alpha<1$.

Answer 1

This is a well-studied problem. The number of zeroes of a Dirichlet $L$-function $L(s,\chi)$ associated to a primitive even Dirichlet character $\chi$ modulo $q > 1$ up to height $T \geq 1$, $$N(T,\chi) = \{\rho = \beta + i\gamma : L(\rho,\chi) = 0, \ \beta \in (0,1), \ |\gamma| \leq T\},$$ can readily be shown (e.g. Theorem 5.8 of Iwaniec-Kowalski) to satisfy $$N(T,\pi) = \frac{1}{\pi} \int_{-T}^{T} \left(-\frac{1}{2} \log \pi + \frac{1}{2} \frac{\Gamma'}{\Gamma}\left(\frac{1}{4} + \frac{it}{2}\right)\right) \, dt + S(\chi,T) + S(\overline{\chi},T).$$ Here $$S(\chi,T) := \frac{1}{\pi} \arg L\left(\frac{1}{2} + it,\chi\right)$$ satisfies $S(\chi,T) = O(\log(q(T + 1)))$. Assuming the generalised Riemann hypothesis, the error term can be improved to be $$O\left(\frac{\log (q(T + 1))}{\log \log (q(T + 1))}\right).$$ (See e.g. https://doi.org/10.1007/s00209-015-1485-9) However we do not know how to improve this further, such as a bound of the form $O(\log^{\alpha}(q(T + 1)))$ for some fixed $\alpha < 1$.

Answer 2

What follows is a refinement of Peter's answer that might be useful to you, depending on what exactly you want to pursue. Here is what is known for $\zeta(s)$ on RH: If $t$ is large and $0<h\leq \sqrt{t}$, then the number of zeros $1/2+i\gamma$ of $\zeta(s)$ with $\gamma\in[t,t+h]$ is

$\frac{h}{2\pi}\log\frac{t}{2\pi} + O\Big(\frac{\log t}{\log\log t}\Big)$.

See Goldston and Gonek. One can modify their approach for Dirichlet $L$-functions to obtain the count

$\frac{h}{2\pi}\log\frac{qt}{2\pi} + O\Big(\frac{\log qt}{\log\log qt}\Big)$.

(Many pertinent details can be found in Carneiro, Chandee, and Milinovich.) Additionally, one can determine the implied constant with relatively high accuracy.

Goldston, D. A.; Gonek, S. M., A note on (S(t)) and the zeros of the Riemann zeta-function, Bull. Lond. Math. Soc. 39, No. 3, 482-486 (2007). ZBL1127.11058.

Carneiro, Emanuel; Chandee, Vorrapan; Milinovich, Micah B., A note on the zeros of zeta and (L)-functions, Math. Z. 281, No. 1-2, 315-332 (2015). ZBL1332.11078.