Moments of the probability density functions of error terms in divisor problems

Lau, Yuk-Kam; Tsang, Kai-Man

doi:10.1090/S0002-9939-04-07825-6

Moments of the probability density functions of error terms in divisor problems
HTML articles powered by AMS MathViewer

by Yuk-Kam Lau and Kai-Man Tsang PDF

Proc. Amer. Math. Soc. 133 (2005), 1283-1290 Request permission

Abstract:

We derive an explicit formula for the moments of the probability density function of a class of functions. An application of this shows that the density function of the error term in the Pilz divisor problem is asymmetric.

References

Pavel M. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems, Duke Math. J. 70 (1993), no. 3, 655–682. MR 1224102, DOI 10.1215/S0012-7094-93-07015-9
Pavel M. Bleher, Zheming Cheng, Freeman J. Dyson, and Joel L. Lebowitz, Distribution of the error term for the number of lattice points inside a shifted circle, Comm. Math. Phys. 154 (1993), no. 3, 433–469. MR 1224087, DOI 10.1007/BF02102104
D. R. Heath-Brown, The distribution and moments of the error term in the Dirichlet divisor problem, Acta Arith. 60 (1992), no. 4, 389–415. MR 1159354, DOI 10.4064/aa-60-4-389-415
Yuk-Kam Lau, On the existence of limiting distributions of some number-theoretic error terms, J. Number Theory 94 (2002), no. 2, 359–374. MR 1916279, DOI 10.1006/jnth.2001.2734
Kai Man Tsang, Higher-power moments of $\Delta (x),\;E(t)$ and $P(x)$, Proc. London Math. Soc. (3) 65 (1992), no. 1, 65–84. MR 1162488, DOI 10.1112/plms/s3-65.1.65
Wenguang Zhai, On higher-power moments of $\Delta (x)$. II, Acta Arith. 114 (2004), no. 1, 35–54. MR 2067871, DOI 10.4064/aa114-1-3