Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

Simonič, Aleksander; Trudgian, Timothy; Turnage-Butterbaugh, Caroline

doi:10.1090/tran/8571

Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function
HTML articles powered by AMS MathViewer

by Aleksander Simonič, Timothy S. Trudgian and Caroline L. Turnage-Butterbaugh PDF

Trans. Amer. Math. Soc. 375 (2022), 3239-3265 Request permission

Abstract:

We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $\zeta (s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $\zeta (s)$ on the critical line and $h \ll 1 / \log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $\zeta (s)$ of a given multiplicity.

References

H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4167–4175. MR 2680043, DOI 10.1090/S0002-9939-2010-10443-4
H. M. Bui and M. B. Milinovich, Gaps between zeros of the Riemann zeta-function, Q. J. Math. 69 (2018), no. 2, 403–423. MR 3811585, DOI 10.1093/qmath/hax047
Andrés Chirre, Felipe Gonçalves, and David de Laat, Pair correlation estimates for the zeros of the zeta function via semidefinite programming, Adv. Math. 361 (2020), 106926, 22. MR 4037496, DOI 10.1016/j.aim.2019.106926
J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16 (1984), no. 4, 421–424. MR 749453, DOI 10.1112/blms/16.4.421
J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown, On the distribution of gaps between zeros of the zeta-function, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 43–51. MR 780348, DOI 10.1093/qmath/36.1.43
J. Brian Conrey and Caroline L. Turnage-Butterbaugh, On $r$-gaps between zeros of the Riemann zeta-function, Bull. Lond. Math. Soc. 50 (2018), no. 2, 349–356. MR 3830125, DOI 10.1112/blms.12142
David W. Farmer, Counting distinct zeros of the Riemann zeta-function, Electron. J. Combin. 2 (1995), Research Paper 1, approx. 5 pp. (electonic). MR 1309123
Shaoji Feng and Xiaosheng Wu, On gaps between zeros of the Riemann zeta-function, J. Number Theory 132 (2012), no. 7, 1385–1397. MR 2903162, DOI 10.1016/j.jnt.2011.12.014
Akio Fujii, On the zeros of Dirichlet $L$-functions. I, Trans. Amer. Math. Soc. 196 (1974), 225–235. MR 349603, DOI 10.1090/S0002-9947-1974-0349603-2
Akio Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals, Bull. Amer. Math. Soc. 81 (1975), 139–142. MR 354575, DOI 10.1090/S0002-9904-1975-13674-3
Akio Fujii, On the difference between $r$ consecutive ordinates of the zeros of the Riemann zeta function, Proc. Japan Acad. 51 (1975), no. 10, 741–743. MR 389781
Akio Fujii, On the zeros of Dirichlet $L$-functions. II, Trans. Amer. Math. Soc. 267 (1981), no. 1, 33–40. MR 621970, DOI 10.1090/S0002-9947-1981-0621970-5
A. Ghosh, On Riemann’s zeta function—sign changes of $S(T)$, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 25–46. MR 637341
R. R. Hall, The behaviour of the Riemann zeta-function on the critical line, Mathematika 46 (1999), no. 2, 281–313. MR 1832621, DOI 10.1112/S0025579300007762
R. R. Hall, A Wirtinger type inequality and the spacing of the zeros of the Riemann zeta-function, J. Number Theory 93 (2002), no. 2, 235–245. MR 1899304, DOI 10.1006/jnth.2001.2719
R. R. Hall, A new unconditional result about large spaces between zeta zeros, Mathematika 52 (2005), no. 1-2, 101–113 (2006). MR 2261847, DOI 10.1112/S0025579300000383
Habiba Kadiri, Allysa Lumley, and Nathan Ng, Explicit zero density for the Riemann zeta function, J. Math. Anal. Appl. 465 (2018), no. 1, 22–46. MR 3806689, DOI 10.1016/j.jmaa.2018.04.071
A. A. Karatsuba and M. A. Korolëv, The argument of the Riemann zeta function, Uspekhi Mat. Nauk 60 (2005), no. 3(363), 41–96 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 3, 433–488. MR 2167812, DOI 10.1070/RM2005v060n03ABEH000848
A. A. Karatsuba and M. A. Korolëv, The behavior of the argument of the Riemann zeta function on the critical line, Russian Math. Surveys 61 (2006), no. 3(369), 389–482.
M. A. Korolëv, On large gaps between consecutive zeros of the Riemann zeta function, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 2, 91–104 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 2, 291–304. MR 2413651, DOI 10.1070/IM2008v072n02ABEH002402
H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Topics in classical number theory, Vol. I, II (Budapest, 1981) Colloq. Math. Soc. János Bolyai, vol. 34, North-Holland, Amsterdam, 1984, pp. 1079–1106. MR 781177
Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to $3\cdot 10^{12}$, Bull. Lond. Math. Soc. 53 (2021), no. 3, 792–797. MR 4275089, DOI 10.1112/blms.12460
Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, and Dirk Zeindler, More than five-twelfths of the zeros of $\zeta$ are on the critical line, Res. Math. Sci. 7 (2020), no. 2, Paper No. 2, 74. MR 4045193, DOI 10.1007/s40687-019-0199-8
E. Preissmann, Sur une inégalité de Montgomery-Vaughan, Enseign. Math. (2) 30 (1984), no. 1-2, 95–113 (French). MR 743672
Sergei Preobrazhenskiĭ, A small improvement in the gaps between consecutive zeros of the Riemann zeta-function, Res. Number Theory 2 (2016), Paper No. 28, 11. MR 3580849, DOI 10.1007/s40993-016-0053-7
O. Ramaré, An explicit density estimate for Dirichlet $L$-series, Math. Comp. 85 (2016), no. 297, 325–356. MR 3404452, DOI 10.1090/mcom/2991
Brad Rodgers and Terence Tao, The de Bruijn–Newman constant is non-negative, Forum Math. Pi 8 (2020), e6, 62. MR 4089393, DOI 10.1017/fmp.2020.6
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Atle Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89–155. MR 20594
Aleksander Simonič, Lehmer pairs and derivatives of Hardy’s $Z$-function, J. Number Theory 184 (2018), 451–460. MR 3724173, DOI 10.1016/j.jnt.2017.08.030
Aleksander Simonič, Explicit zero density estimate for the Riemann zeta-function near the critical line, J. Math. Anal. Appl. 491 (2020), no. 1, 124303, 41. MR 4114203, DOI 10.1016/j.jmaa.2020.124303
K. Soundararajan, On the distribution of gaps between zeros of the Riemann zeta-function, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 187, 383–387. MR 1412563, DOI 10.1093/qmath/47.3.383
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Timothy S. Trudgian, Selberg’s method and the multiplicities of the zeroes of the Riemann zeta-function, Comment. Math. Univ. St. Pauli 60 (2011), no. 1-2, 227–229. MR 2951934
Timothy S. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 134 (2014), 280–292. MR 3111568, DOI 10.1016/j.jnt.2013.07.017
Kai-Man Tsang, THE DISTRIBUTION OF THE VALUES OF THE RIEMANN ZETA-FUNCTION, ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–Princeton University. MR 2633927
Kai Man Tsang, Some $\Omega$-theorems for the Riemann zeta-function, Acta Arith. 46 (1986), no. 4, 369–395. MR 871279, DOI 10.4064/aa-46-4-369-395
XiaoSheng Wu, A note on the distribution of gaps between zeros of the Riemann zeta-function, Proc. Amer. Math. Soc. 142 (2014), no. 3, 851–857. MR 3148519, DOI 10.1090/S0002-9939-2013-11833-2