The mean square discrepancy in the divisor problem

Ge, Fan; Gonek, Steven

doi:10.1090/tran/8087

The mean square discrepancy in the divisor problem
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by Fan Ge and Steven M. Gonek PDF

Trans. Amer. Math. Soc. 373 (2020), 4713-4734 Request permission

Abstract:

We study the mean square of the error term, or the mean square discrepancy, in the Dirichlet divisor problem. An analysis of the off-diagonal terms in the mean square of the Voronoi summation formula leads to a precise conjecture for this mean square discrepancy. Surprisingly, the discrepancy contains a slowly oscillating, unbounded term.

References

Steven M. Gonek and Alex Iosevich, The mean square discrepancy in the circle problem, Mosc. J. Comb. Number Theory 8 (2019), no. 3, 263–287. MR 3990808, DOI 10.2140/moscow.2019.8.263
James Lee Hafner, New omega theorems for two classical lattice point problems, Invent. Math. 63 (1981), no. 2, 181–186. MR 610536, DOI 10.1007/BF01393875
G. H. Hardy, On Dirichlet’s Divisor Problem, Proc. London Math. Soc. (2) 15 (1916), 1–25. MR 1576550, DOI 10.1112/plms/s2-15.1.1
M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609. MR 2005876, DOI 10.1112/S0024611503014485
A. Ivić, The Laplace transform of the square in the circle and divisor problems, Studia Sci. Math. Hungar. 32 (1996), no. 1-2, 181–205. MR 1405136
Yuk-Kam Lau and Kai-Man Tsang, Mean square of the remainder term in the Dirichlet divisor problem, J. Théor. Nombres Bordeaux 7 (1995), no. 1, 75–92. Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993). MR 1413567, DOI 10.5802/jtnb.131
Yuk-Kam Lau and Kai-Man Tsang, On the mean square formula of the error term in the Dirichlet divisor problem, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 277–287. MR 2475967, DOI 10.1017/S0305004108001874
Tom Meurman, On the mean square of the Riemann zeta-function, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 151, 337–343. MR 907241, DOI 10.1093/qmath/38.3.337
Tom Meurman, On the binary additive divisor problem, Number theory (Turku, 1999) de Gruyter, Berlin, 2001, pp. 223–246. MR 1822012
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
Emmanuel Preissmann, Sur la moyenne quadratique du terme de reste du problème du cercle, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 4, 151–154 (French, with English summary). MR 930552
K. Soundararajan, Omega results for the divisor and circle problems, Int. Math. Res. Not. 36 (2003), 1987–1998. MR 1991181, DOI 10.1155/S1073792803130309
Kwang-Chang Tong, On divisor problems. II, III, Acta Math. Sinica 6 (1956), 139–152, 515–541 (Chinese, with English summary). MR 98718
Kai Man Tsang, Mean square of the remainder term in the Dirichlet divisor problem. II, Acta Arith. 71 (1995), no. 3, 279–299. MR 1339133, DOI 10.4064/aa-71-3-279-299
Kai-Man Tsang, Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function, Sci. China Math. 53 (2010), no. 9, 2561–2572. MR 2718848, DOI 10.1007/s11425-010-4068-6
Georges Voronoï, Sur une fonction transcendante et ses applications à la sommation de quelques séries (suite), Ann. Sci. École Norm. Sup. (3) 21 (1904), 459–533 (French). MR 1509047, DOI 10.24033/asens.545