Jaskirat Singh

math.NT — Number Theory

Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant $\pi^2/15$

Contributed by Jaskirat Singh

Let $\gamma_n<\gamma_{n+1}$ be consecutive ordinates of zeros of $\zeta(\tfrac12+it)$ with normalized gap $s_n$. We study the derivatives $Z'(\gamma_n)$, $Z'(\gamma_{n+1})$ of Hardy's function at close pairs. For the circular unitary ensemble we prove that the conditional variance of $\log|\Lambda'(\theta_1)|-\log|\Lambda'(\theta_2)|$ at adjacent eigenangles with normalized gap $s$ satisfies $\operatorname{Var}=c_N s^2(1+o(1))$ as $s\to0$, with the exact value $c_N=\frac{\pi^2}{15}\bigl(1-\frac{4}{N^2}\bigr)$, so that $c_\infty=\pi^2/15$, which we call the Singh constant. The proof reduces the problem through a Cayley transform to the Hua ensemble and closes it with two loop equations. We state a conditional theorem transferring the constant to the zeta function under the Riemann Hypothesis, a local GUE hypothesis and a tail bound, and we formulate a conjecture on the size of $Z'$ at close pairs. Numerical evidence from Odlyzko's tables up to zero number $10^{22}$ agrees with the predictions. This paper is a revised and expanded version of the Zenodo record doi:10.5281/zenodo.23899228.