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1

The sum of divisors of a quadratic form

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Zhao L.

Acta Arithmetica

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2014

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tom 163

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nr 2

161-177

EN

We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let $S₃(X) = ∑_{1≤m₁,m₂,m₃≤X} τ(m₁²+m₂²+m₃²)$. We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_ε(X^{8/3+ε})$ obtained by Guo and Zhai (2012).

2

Exceptional sets in Waring's problem: two squares and s biquadrates

100%

Zhao L.

Acta Arithmetica

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2014

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tom 162

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nr 4

369-379

EN

Let $R_s(n)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s=3$ or 4, it is established that the anticipated asymptotic formula for $R_s(n)$ holds for all $n≤X$ with at most $O(X^{(9-2s)/8+ε })$ exceptions.

3

Four squares of primes and powers of 2

100%

Zhao L.

Acta Arithmetica

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2014

|

tom 162

|

nr 3

255-271

EN

By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.

Ograniczanie wyników

3 Acta Arithmetica

3 Zhao L.

3 2014

The sum of divisors of a quadratic form

Exceptional sets in Waring's problem: two squares and s biquadrates

Four squares of primes and powers of 2