On differences of two squares

• Autorzy: Manfred Kühleitner , Werner Nowak
• Języki publikacji: EN

Abstrakty

EN

The arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

Słowa kluczowe

EN

Arithmetic functions; divisor problems; lattice points; 11N37

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 1
• Strony: 110-122

Daty

wydano

2006-03-01

online

2006-03-01

Twórcy

autor

Manfred Kühleitner

• Universität für Bodenkultur Wien

autor

Werner Nowak

• Universität für Bodenkultur Wien

Bibliografia

• [1] K. Corrádi and I. Kátai: “A comment on K.S. Gangadharan's paper “Two classical lattice point problems”” (in Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., Vol. 17, (1967), pp. 89–97.
• [2] J.L. Hafner: “New omega theorems for two classical lattice point problems”, Invent. Math., Vol. 63, (1981), pp. 181–186. http://dx.doi.org/10.1007/BF01393875
• [3] J.L. Hafner: “On the average order of a class of arithmetical functions”, J. Number Theory, Vol. 15, (1982), pp. 36–76. http://dx.doi.org/10.1016/0022-314X(82)90082-8
• [4] M. Huxley: “Exponential sums and lattice points III”, Proc. London Math. Soc., Vol. 87(3), (2003), pp. 591–609. http://dx.doi.org/10.1112/S0024611503014485
• [5] A. Ivić, The Riemann zeta-function, Wiley & Sons, New York 1985.
• [6] A. Ivić, E. Krätzel, M. Kühleitner and W.G. Nowak: “Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic”, In: W. Schwarz (Ed.): Proc. Conf. on Elementary and Analytic Number Theory ELAZ'04, held in Mainz, to appear; Available in electronic form at http://arXiv.org/pdf/math.NT/0410522.
• [7] D.R. Heath-Brown: “The distribution and moments of the error term in the Dirichlet divisor problems”, Acta Arith., Vol. 60, (1992), pp. 389–415.
• [8] E. Krätzel: Lattice points, Kluwer, Dordrecht-Boston-London, 1988.
• [9] E. Krätzel: Analytische Funktionen in der Zahlentheorie, Teubner, Stuttgart-Leipzig-Wiesbaden, 2000.
• [10] M. Kühleitner: “An Omega theorem on differences of two squares”, Acta Math. Univ. Comen., New Ser., Vol. 61, (1992), pp. 117–123.
• [11] M. Kühleitner: “An Omega theorem on differences of two squares, II”, Acta Math. Univ. Comen., New Ser., Vol. 68, (1999), pp. 27–35.
• [12] Y.-K. Lau and K.-M. Tsang: “Omega result for the mean square of the Riemann zeta function”, Manuscr. Math., Vol. 117, (2005), pp. 373–381. http://dx.doi.org/10.1007/s00229-005-0565-2
• [13] Y.-K. Lau and K.-M. Tsang: “Moments over short intervals”, Arch. Math., Vol. 84, (2005), pp. 249–257. http://dx.doi.org/10.1007/s00013-004-1119-7
• [14] T. Meurman: “On the mean square of the Riemann zeta-function”, Quart. J. Math. Oxford, Vol. 38(2), (1987), pp. 337–343.
• [15] H.L. Montgomery and R.C. Vaughan: “Hilbert's inequality”, J. London, Vol. 8(2), (1974), pp. 73–82.
• [16] W.G. Nowak: “On the divisor problem: Moments of Δ(x) over short intervals”, Acta Arithm., Vol. 109, (2003), pp. 329–341.
• [17] E. Preissmann: “Sur la moyenne quadratique du terme de reste du problème du cercle”, C. R. Acad. Sci., Paris, Sér. I, Vol. 306, (1988), pp. 151–154.
• [18] K. Soundararajan: “Omega results for the divisor and circle problems”, Int. Math. Res. Not., Vol. 36, (2003), pp. 1987–1998. http://dx.doi.org/10.1155/S1073792803130309
• [19] E.C. Titchmarsh: The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951.
• [20] K.C. Tong: “On divisor problems”, Acta Math. Sinica, Vol. 6, (1956), pp. 515–541.
• [21] K.-M. Tsang: “Higher power moments of Δ(x), E(t), and P(x)”, Proc. London Math. Soc., III. Ser., Vol. 65, (1992), pp. 65–84.
• [22] W. Zhai: “On higher-power moments of Δ(x)”, Acta Arith., Vol. 112, (2004), pp. 367–395.
• [23] W. Zhai: “On higher-power moments of Δ(x), II”, Acta Arith., Vol. 114, (2004), pp. 35–54. http://dx.doi.org/10.4064/aa114-1-3
• [24] W. Zhai: “On higher-power moments of Δ(x), III”, Acta Arith., Vol. 118, (2005), pp. 263–281.

Identyfikatory

DOI

10.1007/s11533-005-0007-0