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

Jaskirat Singh

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

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Contributed by Jaskirat Singh ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

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.

References

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Show all 12 referencesShow fewer references

J. Singh, Zenodo record, doi:10.5281/zenodo.23899228 (https://doi.org/10.5281/zenodo.23899228).DOI
A. M. Turing, Some calculations of the Riemann zeta function , Proc. London Math. Soc. (3) 3 (1953), 99--117.

Version history

v1 · 7 pagesSubmitted by Jaskirat SinghInitial depositCurrentNov 03, 2026