A note on the distribution of integer points on spheres

作者：Luo Wenzhi^*

来源：Mathematische Zeitschrift, 2011, 267(3-4): 965-970.

DOI：10.1007/s00209-009-0656-y

• Equidistribution
• Spherical harmonic
• Quarternion algebra
• Jacquet-Langlands correspondence

In this note, by applying the works of Bocherer and Schulze-Pillot, we evaluate asymptotically the variance of the Linnik distribution, and show as Y -> infinity, Sigma(n <= Y), (n not equivalent to 0(mod 4)) Sigma (2)(vertical bar x vertical bar) =n P (x/vertical bar x vertical bar)/ n(1/4) Sigma(vertical bar z vertical bar)2 =n Q (z/vertical bar z vertical bar)/n(1/4) = c < P, Q > Delta (1/2, P) Y + 0 (Y), for any spherical harmonics P, Q on S (2) which are Hecke eigenforms, where | center dot | is the usual Euclidean norm on R (3), and c is an explicit constant.

• 出版日期2011-4