{"id":"hexagon/2611.00059","pdf":"https://hexagonmath.org/pdf/2611.00059v1","url":"https://hexagonmath.org/abs/2611.00059v1","tags":["math.PR","math-ph","msc:11M26","msc:15B52","msc:60B20","Hardy Z-function","Riemann Zeta Function","circular unitary ensemble","random matrix theory","zero spacing"],"score":0,"title":"Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant $\\pi^2/15$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Jaskirat Singh"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}