## Tsukuba Journal of Mathematics

### Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations

#### Abstract

Let $R_{s,k}(n)$ denote the number of representations of $n$ as the sum of $s$ positive integral $k$th powers. We develop methods to establish asymptotic formulae for moments of $R_{s,k}(n)$ in which $n$ is restricted to a thin sequence consisting of values of a given polynomial. As a first example, we discuss the case of cubes, and present conclusions for sums of 6 and 7 cubes in which the polynomial is quadratic, and for sums of 7 cubes in which the polynomial is cubic. We also consider the case wherein the exponent $k$ is large, and briefly describe corresponding results for the binary Goldbach problem.

#### Article information

Source
Tsukuba J. Math., Volume 32, Number 2 (2008), 383-406.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496165237

Digital Object Identifier
doi:10.21099/tkbjm/1496165237

Mathematical Reviews number (MathSciNet)
MR2477988

Zentralblatt MATH identifier
1183.11059

#### Citation

Brüdern, Jörg; Kawada, Koichi; Wooley, Trevor D. Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations. Tsukuba J. Math. 32 (2008), no. 2, 383--406. doi:10.21099/tkbjm/1496165237. https://projecteuclid.org/euclid.tkbjm/1496165237