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Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant π2/15\pi^2/15

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.PR — Probabilitymath-ph — Mathematical Physics

Contributed by Jaskirat Singh

Version 2 / Nov 05, 2026 / CC BY 4.0

Abstract

Let γn<γn+1\gamma_n<\gamma_{n+1} be consecutive ordinates of zeros of ζ(12+it)\zeta(\tfrac12+it) with normalized gap sns_n. We study the derivatives Z′(γn)Z'(\gamma_n), Z′(γn+1)Z'(\gamma_{n+1}) of Hardy's function at close pairs. For the circular unitary ensemble we prove that the conditional variance of log⁡∣Λ′(θ1)∣−log⁡∣Λ′(θ2)∣\log|\Lambda'(\theta_1)|-\log|\Lambda'(\theta_2)| at adjacent eigenangles with normalized gap ss satisfies Var⁡=cNs2(1+o(1))\operatorname{Var}=c_N s^2(1+o(1)) as s→0s\to0, with the exact value cN=π215(1−4N2)c_N=\frac{\pi^2}{15}\bigl(1-\frac{4}{N^2}\bigr), so that c∞=π2/15c_\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′Z' at close pairs. Numerical evidence from Odlyzko's tables up to zero number 102210^{22} agrees with the predictions. This paper is a revised and expanded version of the Zenodo record doi:10.5281/zenodo.23899228.

Author comments: v2: Section 1 now lists fourteen Singh Lemmas, each labeled proved (6, 8, 45), conditional (7, 27, 42, 44, 46, 48, 54, 67) or numerical (5, 51, 57). The unitary pi^2/15 constant and Hua moments are credited as known results, Lemma 20 is credited to Soundararajan and Milinovich, and method credits and an explicit transfer hypothesis for Lemma 67 are added.

References

  1. E. Bogomolny and J. P. Keating, A method for calculating spectral statistics based on random matrix universality with an application to the three point correlations of the Riemann zeros , J. Phys. A 46 (2013), 305203.
  2. H. M. Bui and D. R. Heath-Brown, On simple zeros of the Riemann zeta-function , Bull. Lond. Math. Soc. 45 (2013), 953-961.
  3. E. Carneiro, V. Chandee, F. Littmann and M. B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function , J. Reine Angew. Math. 725 (2017), 143-182; arXiv:1406.5462.arXiv
  4. A. Chirre, F. Gon ç alves and D. de Laat, Fourier optimization and small gaps in the zeros of the Riemann zeta function , preprint; positive proportion of gaps ≤0.6039\le 0.6039 of the mean.
  5. A. Chirre, F. Gon ç alves and D. de Laat, Pair correlation estimates for the zeros of the zeta function via semidefinite programming , Adv. Math. 361 (2020), 106926; arXiv:1810.08843.arXiv
  6. F. J. Dyson, Statistical theory of the energy levels of complex systems. I , J. Math. Phys. 3 (1962), 140-156.
  7. D. A. Goldston, S. M. Gonek and H. L. Montgomery, Mean values of the logarithmic derivative of the Riemann zeta function with applications to primes in short intervals , J. Reine Angew. Math. 537 (2001), 105-122.
  8. D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals , in Analytic Number Theory and Diophantine Problems (Stillwater, OK, 1984), Progr. Math. 70 , Birkhäuser, Boston, 1987, 183-203; doi:10.1007/978-1-4612-4816-3_10.DOI
  9. D. A. Goldston and R. C. Vaughan, On the Montgomery-Hooley asymptotic formula , in Sieve Methods, Exponential Sums, and their Applications in Number Theory (Cardiff, 1995), London Math. Soc. Lecture Note Ser. 237 , Cambridge Univ. Press, Cambridge, 1997, 117-142; doi:10.1017/cbo9780511526091.010.DOI
  10. S. M. Gonek, Mean values of the Riemann zeta-function and its derivatives , Invent. Math. 75 (1984), 123-141; subsequent uniform Landau formulae in S. M. Gonek, An explicit formula of Landau and its applications , Contemp. Math. 143 (1993), 395-406.
Show all 25 referencesShow fewer references
  1. D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity , Canad. J. Math. 34 (1982), 1365-1377; doi:10.4153/CJM-1982-095-9.DOI
  2. C. Hooley, On the Barban-Davenport-Halberstam theorem. I , J. Reine Angew. Math. 274/275 (1975), 206-223; doi:10.1515/crll.1975.274-275.206.DOI
  3. L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains , Amer. Math. Soc., Providence, 1963.
  4. C. P. Hughes, J. P. Keating and N. O'Connell, Random matrix theory and the derivative of the Riemann zeta function , Proc. R. Soc. Lond. A 456 (2000), 2611-2627.
  5. J. P. Keating and N. C. Snaith, Random matrix theory and ζ(1/2+it)\zeta(1/2+it) , Comm. Math. Phys. 214 (2000), 57-89.
  6. M. L. Mehta, Random Matrices , 3rd ed., Pure Appl. Math. 142 , Elsevier/Academic Press, Amsterdam, 2004.
  7. M. B. Milinovich, Upper bounds for moments of ζ′(ρ)\zeta'(\rho) , Bull. Lond. Math. Soc. 42 (2010), 28-44; arXiv:0806.0786.arXiv
  8. H. L. Montgomery, The pair correlation of zeros of the zeta function , Proc. Sympos. Pure Math. 24 (1973), 181-193.
  9. A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function , Math. Comp. 48 (1987), 273-308.
  10. A. M. Odlyzko, The 102010^{20}-th zero of the Riemann zeta function and 175 million of its neighbors , preprint, AT&T Bell Laboratories, 1992.
  11. A. M. Odlyzko, The 102210^{22}-nd zero of the Riemann zeta function , in Dynamical, Spectral, and Arithmetic Zeta Functions, Contemp. Math. 290 , Amer. Math. Soc., Providence, 2001, 139-144.
  12. Z. Rudnick and P. Sarnak, Zeros of principal LL-functions and random matrix theory , Duke Math. J. 81 (1996), 269-322; doi:10.1215/S0012-7094-96-08115-6.DOI
  13. J. Singh, companion data and notes, Zenodo, doi:10.5281/zenodo.23899228.DOI
  14. K. Soundararajan, Moments of the Riemann zeta function , Ann. of Math. (2) 170 (2009), 981-993; arXiv:math/0612106.arXiv
  15. A. M. Turing, Some calculations of the Riemann zeta-function , Proc. London Math. Soc. (3) 3 (1953), 99-117.

Version history

  1. v1 · 7 pagesSubmitted by Jaskirat SinghInitial depositSupersededNov 03, 2026
  2. v2 · 9 pagesSubmitted by Jaskirat Singhv2: Section 1 now lists fourteen Singh Lemmas, each labeled proved (6, 8, 45), conditional (7, 27, 42, 44, 46, 48, 54, 67) or numerical (5, 51, 57). The unitary pi^2/15 constant and Hua moments are credited as known results, Lemma 20 is credited to Soundararajan and Milinovich, and method credits and an explicit transfer hypothesis for Lemma 67 are added.CurrentNov 05, 2026