RUS  ENG
Full version
JOURNALS // Uspekhi Matematicheskikh Nauk

Uspekhi Mat. Nauk, 2024, Volume 79, Issue 1(475), Pages 59–134 (Mi rm10162)

Voronoi's formulae and the Gauss problem
D. A. Popov

References

1. E. Landau, Vorlesungen über Zahlentheorie, v. 2, Hierzel, Leipzig, 1927, viii+308 pp.  mathscinet  zmath
2. E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1951, vi+346 pp.  mathscinet  zmath
3. A. Ivić, The Rieman zeta-function. The theory of the Riemann zeta-function with applications, Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York, 1985, xvi+517 pp.  mathscinet  zmath
4. A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin, 1993, xiv+222 pp.  crossref  mathscinet  mathscinet  zmath  zmath
5. E. Krätzel, Lattice points, Math. Appl. (East European Ser.), 33, Kluwer Acad. Publ., Dordrecht, 1988, 320 pp.  mathscinet  zmath
6. Kai-Man Tsang, “Recent progress on the Dirichlet divisor problem and the mean square of Riemann zeta-function”, Sci. China Math., 53:9 (2010), 2561–2572  crossref  mathscinet  zmath
7. D. A. Popov, “Circle problem and the spectrum of the Laplace operator on closed 2-manifolds”, Russian Math. Surveys, 74:5 (2019), 909–925  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
8. A. G. Postnikov, Introduction to analytic number theory, Transl. Math. Monogr., 68, Amer. Math. Soc., Providence, RI, 1988, vi+320 pp.  crossref  mathscinet  mathscinet  zmath  zmath
9. E. Bombieri, H. Iwaniec, “On the order of $\zeta(\frac{1}{2}+it)$”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13:3 (1986), 449–472  mathscinet  zmath
10. S. W. Graham, G. Kolesnik, Van der Corput's method of exponential sums, London Math. Soc. Lecture Note Ser., 126, Cambridge Univ. Press, Cambridge, 1991, vi+120 pp.  crossref  mathscinet  zmath
11. M. N. Huxley, Area, lattice points, and exponential sums, London Math. Soc. Monogr. (N. S.), 13, The Clarendon Press, Oxford Univ. Press, New York, 1996, xii+494 pp.  mathscinet  zmath
12. G. Kolesnik, “On the method of exponential pairs”, Acta Arith., 45:2 (1985), 115–143  crossref  mathscinet  zmath
13. H. Iwaniec, C. J. Mozzochi, “On the divisor and circle problems”, J. Number Theory, 29:1 (1988), 60–93  crossref  mathscinet  zmath
14. Xiaochun Li, Xuerui Yang, An improvement on Gauss's circle problem and Dirichlet's divisor problem, 2023, 32 pp., arXiv: 2308.14859v1
15. G. Voronoï, “Sur le développement, à l'aide des fonctions cylindriques, des sommes doubles $\sum f(pm^2+2qmn+rn^2)$, où $pm^2+2qmn+rn^2$ est une forme positive à coefficients entiers”, Verhandlungen des dritten internationalen Mathematiker-Kongresses (Heidenberg, 1904), Teubner, Leipzig, 1905, 241–245  zmath
16. G. H. Hardy, “On the expression of number as the sum of two squares”, Quat. J. Pure Appl. Math., 46 (1915), 263–283  zmath
17. K. F. Ireland, M. I. Rosen, A classical introduction to modern number theory, Grad. Texts in Math., 84, Springer-Verlag, New York–Berlin, 1982, xiii+341 pp.  mathscinet  mathscinet  zmath  zmath
18. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verlagsges., Leipzig, 1923, viii+265 pp.  mathscinet  zmath
19. S. Bochner, Lectures on Fourier integrals, With an author's supplement on monotonic functions, Stieltjes integrals, and harmonic analysis, Ann. of Math. Stud., 42, Princeton Univ. Press, Princeton, NJ, 1959, viii+333 pp.  crossref  mathscinet  zmath  zmath
20. G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge, England; The Macmillan Co., New York, 1944, vi+804 pp.  mathscinet  zmath
21. E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, Clarendon Press, 1937, x+390 pp.  mathscinet  zmath
22. G. H. Hardy, M. Reisz, The general theory of Dirichlet's series, Cambridge Tracts in Math. and Math. Phys., 18, Cambridge Univ. Press, Cambridge, 1964, vii+78 pp.  mathscinet  zmath
23. D. A. Popov, “Spectrum of the Laplace operator on closed surfaces”, Russian Math. Surveys, 77:1 (2022), 81–97  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
24. G. H. Hardy, E. Landau, “The lattice points of a circle”, Proc. Roy. Soc. London Ser. A, 105:731 (1924), 244–258  crossref  zmath
25. K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1957, x+415 pp.  mathscinet  mathscinet  zmath  zmath
26. K. Chandrasekharan, Arithmetical functions, Grundlehren Math. Wiss., 167, Springer-Verlag, New York–Berlin, 1970, xi+231 pp.  crossref  mathscinet  mathscinet  zmath  zmath
27. M. Abramowitz, I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964, xiv+1046 pp.  mathscinet  mathscinet  zmath  zmath
28. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, Based, in part, on notes left by H. Bateman, v. 2, McGraw-Hill Book Company, Inc., New York–Toronto–London, 1953, xvii+396 pp.  mathscinet  mathscinet  zmath  zmath  adsnasa
29. I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007, xlviii+1171 pp.  mathscinet  zmath
30. W. G. Nowak, “Lattice points of a circle: an improved mean-square asymptotics”, Acta Arith., 113:3 (2004), 259–272  crossref  mathscinet  zmath
31. Yuk-Kam Lau, Kai-Man Tsang, “On the mean square formula of the error term in the Dirichlet divisor problem”, Math. Proc. Cambridge Philos. Soc., 146:2 (2009), 277–287  crossref  mathscinet  zmath  adsnasa
32. H. L. Montgomery, R. C. Vaughan, “Hilbert's inequality”, J. London Math. Soc. (2), 8 (1974), 73–82  crossref  mathscinet  zmath
33. Kai-Man Tsang, “Higher-power moments of $\Delta(x)$, $E(t)$ and $P(x)$”, Proc. London Math. Soc. (3), 65:1 (1992), 65–84  crossref  mathscinet  zmath
34. Wenguang Zhai, “On higher-power moments of $\Delta(x)$”, Acta Arith., 112:4 (2004), 367–395  crossref  mathscinet  zmath; II, 114:1 (2004), 35–54  crossref  mathscinet  zmath; III, 118:3 (2005), 263–281  crossref  mathscinet  zmath
35. A. Ivić, “Large values of the error term in the divisor problem”, Invent. Math., 71:3 (1983), 513–520  crossref  mathscinet  zmath  adsnasa
36. D. R. Heath-Brown, “The distribution and moments of the error term in the Dirichlet divisor problem”, Acta Arith., 60:4 (1992), 389–415  crossref  mathscinet  zmath
37. G. H. Hardy, “On Dirichlet's divisor problem”, Proc. London Math. Soc. (2), 15 (1916), 1–25  crossref  mathscinet  zmath
38. G. H. Hardy, “The average order of the arithmetical functions $P(x)$ and $\Delta(x)$”, Proc. London Math. Soc. (2), 15 (1916), 192–213  crossref  mathscinet  zmath
39. K. S. Gangadharan, “Two classical lattice point problems”, Proc. Cambridge Philos. Soc., 57:4 (1961), 699–721  crossref  mathscinet  zmath
40. S. Soundararajan, “Omega results for the divisor and circle problems”, Int. Math. Res. Not., 2003:36 (2003), 1987–1998  crossref  mathscinet  zmath
41. D. R. Heath-Brown, K. Tsang, “Sign changes of $E(t)$, $\Delta(x)$, and $P(x)$”, J. Number Theory, 49:1 (1994), 73–83  crossref  mathscinet  zmath
42. M. Kac, Statistical independence in probability, analysis and number theory, Carus Math. Monogr., 12, John Wiley and Sons, Inc., New York, 1959, xiv+93 pp.  mathscinet  zmath  zmath
43. Yuk-Kam Lau, Kai-Man Tsang, “Moments over short intervals”, Arch. Math. (Basel), 84:3 (2005), 249–257  crossref  mathscinet  zmath
44. P. M. Bleher, Zheming Cheng, F. J. Dyson, J. L. Lebowitz, “Distribution of the error term for the number of lattice points inside a shifted circle”, Comm. Math. Phys., 154:3 (1993), 433–469  crossref  mathscinet  zmath  adsnasa
45. Yuk-Kam Lau, “On the tails of the limiting distribution function of the error term in the Dirichlet divisor problem”, Acta Arith., 100:4 (2001), 329–337  crossref  mathscinet  zmath
46. D. A. Popov, “Bounds and behaviour of the quantities $P(x)$ and $\Delta(x)$ on short intervals”, Izv. Math., 80:6 (2016), 1213–1230  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
47. A. Ivić, P. Sargos, “On the higher moments of the error term in the divisor problem”, Illinois J. Math., 51:2 (2007), 353–377  crossref  mathscinet  zmath
48. L. Hörmander, The analysis of linear partial differential operators, v. I, Grundlehren Math. Wiss., 256, Distribution theory and Fourier analysis, Springer-Verlag, Berlin, 1983, ix+391 pp.  crossref  mathscinet  mathscinet  zmath  zmath
49. O. Robert, P. Sargos, “Three-dimensional exponential sums with monomials”, J. Reine Angew. Math., 2006:591 (2006), 1–20  crossref  mathscinet  zmath
50. M. Jutila, “On the divisor problem for short intervals”, Ann. Univ. Turku. Ser. A I, 1984, no. 186, 23–30  mathscinet  zmath
51. A. Ivić, Wenguang Zhai, “On the Dirichlet divisor problem in short intervals”, Ramanujan J., 33:3 (2014), 447–465  crossref  mathscinet  zmath
52. M. A. Korolev, D. A. Popov, “On Jutila's integral in the circle problem”, Izv. Math., 86:3 (2022), 413–455  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
53. A. Ivić, “On the divisor function and the Riemann zeta-function in short intervals”, Ramanujan J., 19:2 (2009), 207–224  crossref  mathscinet  zmath
54. D. A. Popov, D. V. Sushko, “Numerical investigation of the properties of remainder in Gauss's circle problem”, Comput. Math. Math. Phys., 62:12 (2022), 2008–2022  mathnet  crossref  crossref  mathscinet  zmath  adsnasa


© Steklov Math. Inst. of RAS, 2024