On the number variance of zeta zeros and a conjecture of Berry. (English) Zbl 07740600
Let \(\zeta(s)\) be the Riemann zeta function and
\[
S(t)=\frac{1}{\pi}\arg \zeta(\frac{1}{2}+it).
\]
It is expected that there are about \(\delta\) zeros of \(\zeta(s)\) with ordinates in the interval \([t, t+ \frac{2\pi \delta}{\log T}]\) when \(0< t\leq T\) and \(T\) is large. The number variance of the zeros of \(\zeta(s)\) is roughly equal
\[
\int_0^T \left \lbrack S\left(t+ \frac{2\pi \delta}{\log T}\right)-S(t) \right\rbrack ^2 dt
\]
In this paper, the authors introduce new ideas to prove novel results on the number variance of zeta zeros in the non-universal regime when \(\delta \gg \log T\). In particular, they give three different formulations of the above integral and they show how their results give a conditional proof of M. V. Berry’s conjecture [Nonlinearity 1, No. 3, 399–407 (1988; Zbl 0664.10022)] in the non-universal regime assuming RH and a conjecture of Chan for the pair correlation of zeta zeros in longer ranges [T. H. Chan, Acta Arith. 115, No. 2, 181–204 (2004; Zbl 1059.11049)].
Finally, they calculate lower order terms in the second moment of \(\log |\zeta(\frac{1}{2}+it)|\). Assuming Montgomery’s pair correlation conjecture, this establishes a special case of a conjecture of J. P. Keating and N. C. Snaith [Commun. Math. Phys. 214, No. 1, 57–89 (2000; Zbl 1051.11048)].
Finally, they calculate lower order terms in the second moment of \(\log |\zeta(\frac{1}{2}+it)|\). Assuming Montgomery’s pair correlation conjecture, this establishes a special case of a conjecture of J. P. Keating and N. C. Snaith [Commun. Math. Phys. 214, No. 1, 57–89 (2000; Zbl 1051.11048)].
Reviewer: Sami Omar (Sukhair)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
References:
| [1] | M. V.Berry, Semiclassical formula for the number variance of the Riemann zeros, Nonlinearity1 (1988), no. 3, 399-407. · Zbl 0664.10022 |
| [2] | M. V.Berry and J. P.Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev.41 (1999), no. 2, 236-266. · Zbl 0928.11036 |
| [3] | E. B.Bogomolny and J. P.Keating, Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation, Phys. Rev. Lett.77 (1996), no. 8, 1472-1475. · Zbl 1182.81034 |
| [4] | H. M.Bui, S.Lester, and M. B.Milinovich, On Balazard, Saias, and Yor’s equivalence to the Riemann hypothesis, J. Math. Anal. Appl.409 (2014), no. 1, 244-253. · Zbl 1308.11074 |
| [5] | T. H.Chan, Pair correlation of the zeros of the Riemann zeta function in longer ranges, Acta Arith.115 (2004), no. 2, 181-204. · Zbl 1059.11049 |
| [6] | T. H.Chan, Lower order terms of the second moment of \(S(t)\), Acta Arith.123 (2006), no. 4, 313-333. · Zbl 1182.11037 |
| [7] | T. H.Chan, On the second moment of \(S(T)\) in the theory of the Riemann zeta function, Publ. Math.68 (2006), no. 3‐4, 309-329. · Zbl 1110.11028 |
| [8] | A.Chirre and O. E.Quesada‐Herrera, The second moment of \(S_n(t)\) on the Riemann hypothesis, Int. J. Number Theory18 (2022), no. 6, 1203-1226. · Zbl 1489.11117 |
| [9] | J. B.Conrey and N. C.Snaith, Applications of the L‐functions ratios conjectures, Proc. Lond. Math. Soc. (3)94 (2007), no. 3, 594-646. · Zbl 1183.11050 |
| [10] | H.Davenport, Multiplicative number theory, vol. 74, 2nd ed., Springer, 1980. · Zbl 0453.10002 |
| [11] | A.Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals, Bull. Am. Math. Soc.81 (1975), 139-142. · Zbl 0297.10026 |
| [12] | A.Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals, Proc. Japan Acad., Ser. A66 (1990), no. 3, 75-79. · Zbl 0705.11048 |
| [13] | A.Fujii, On the Berry conjecture, J. Math. Kyoto Univ.37 (1997), no. 1, 55-98. · Zbl 0918.11050 |
| [14] | P. X.Gallagher and J. H.Mueller, Primes and zeros in short intervals, J. reine angew. Math.303/304 (1978), 205-220. · Zbl 0396.10028 |
| [15] | D. A.Goldston, On the function \(S(T)\) in the theory of the Riemann zeta‐function, J. Number Theory27 (1987), no. 2, 149-177. · Zbl 0618.10037 |
| [16] | D. A.Goldston and H. L.Montgomery, Pair correlation of zeros and primes in short intervals, Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progress in Mathematics, vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 183-203. · Zbl 0629.10032 |
| [17] | W.Heap, Conditional mean values of long Dirichlet polynomials, Preprint, https://arxiv.org/abs/2201.02108. |
| [18] | J. P.Keating and N. C.Snaith, Random matrix theory and \({\zeta }(1/2+it)\), Commun. Math. Phys.214 (2000), no. 1, 57-89. · Zbl 1051.11048 |
| [19] | J. P.Keating and N. C.Snaith, Random matrix theory and L‐functions at \(s=1/2\), Commun. Math. Phys.214 (2000), no. 1, 91-110. · Zbl 1051.11047 |
| [20] | H. L.Montgomery, The pair correlation of zeros of the zeta function, Proc. Symp. Pure Math.24 (1973), 181-193. · Zbl 0268.10023 |
| [21] | H. L.Montgomery and R. C.Vaughan, Hilbert’s inequality, J. Lond. Math. Soc. (2)8 (1974), 73-82. · Zbl 0281.10021 |
| [22] | H. L.Montgomery and R. C.Vaughan, Multiplicative number theory: I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2006. |
| [23] | A. M.Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comput.48 (1987), 273-308. · Zbl 0615.10049 |
| [24] | M.Radziwiłł and K.Soundararajan, Selberg’s central limit theorem for \(\log \vert{\zeta }(1/2+it)\vert \), Enseign. Math. (2)63 (2017), no. 1-2, 1-19. · Zbl 1432.11117 |
| [25] | A.Selberg, On the remainder in the formula for \(N(T)\), the number of zeros of \({\zeta }(s)\) in the strip \(0<t<T\), Avh. Norske Vid. Akad. Oslo I1944 (1944), no. 1, 1-27. · Zbl 0061.08401 |
| [26] | A.Selberg, Contributions to the theory of the Riemann zeta‐function, Arch. Math. Naturvid.48 (1946), no. 5, 89-155. · Zbl 0061.08402 |
| [27] | A.Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amalfi Conf. Anal. Number Theory (Maiori, 1989), vol. 2, 1992, pp. 47-63. |
| [28] | E. C.Titchmarsh, On the remainder in the formula for \(N(T)\), the number of zeros of \({\zeta }(s)\) in the strip \(0<t<T\), Proc. Lond. Math. Soc. (2)27 (1928), 449-458. · JFM 54.0368.01 |
| [29] | K. M.Tsang, The distribution of the values of the Riemann zeta function, PhD thesis, Princeton University, 1984. |
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