An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas

An asymptotic bound on integer sums of squares

Let σZ(k) be the smallest n such that there exists an identity (x1 + x 2 2 + · · ·+ xk) · (y 1 + y 2 + · · ·+ y k) = f 1 + f 2 + · · ·+ f n , with f1, . . . fn being polynomials with integer coefficients in the variables x1, . . . , xk and y1, . . . , yk. We prove that σZ(k) ≥ Ω(k).

Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids

We express the number of distinct primitive cuboids with given odd diagonal in terms of the twisted Euler function with alternating Dirichlet character of period four, and two counting formulas for binary sums of squares. Based on the asymptotic behaviour of the sums of these formulas, we derive an approximation formula for the cumulative number of primitive cuboids.

Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables

Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...

Explicit Bounds for Sums of Squares

For an even integer k, let r2k(n) be the number of representations of n as a sum of 2k squares. The quantity r2k(n) is approximated by the classical singular series ρ2k(n) nk−1. Deligne’s bound on the Fourier coefficients of Hecke eigenforms gives that r2k(n) = ρ2k(n) + O(d(n)n k−1 2 ). We determine the optimal implied constant in this estimate provided that either k/2 or n is odd. The proof re...

A Short Proof of Milne’s Formulas for Sums of Integer Squares