## 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

## Additional Information

**Aleksander Simonič**- Affiliation: School of Science, The University of New South Wales (Canberra), ACT, Australia
- ORCID: 0000-0003-1298-9031
- Email: a.simonic@student.adfa.edu.au
**Timothy S. Trudgian**- Affiliation: School of Science, The University of New South Wales (Canberra), ACT, Australia
- MR Author ID: 909247
- Email: t.trudgian@adfa.edu.au
**Caroline L. Turnage-Butterbaugh**- Affiliation: Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota
- MR Author ID: 1030621
- ORCID: 0000-0002-5508-1392
- Email: cturnageb@carleton.edu
- Received by editor(s): November 8, 2020
- Received by editor(s) in revised form: September 26, 2021
- Published electronically: December 21, 2021
- Additional Notes: The second author was supported by ARC DP160100932 and FT160100094; the third author was partially supported by NSF DMS-1901293 and NSF DMS-1854398 FRG
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3239-3265 - MSC (2020): Primary 11M06, 11M26; Secondary 11Y35
- DOI: https://doi.org/10.1090/tran/8571
- MathSciNet review: 4402660