Sums of Higher Divisor Functions with Almost Equal Variables

Miao Lou

Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 603 -615.

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Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 603 -615. DOI: 10.1007/s11464-023-0038-5
Research Article

Sums of Higher Divisor Functions with Almost Equal Variables

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Abstract

Let k, ℓ ≥ 3 be integers, and let τk(n) denote the k-th divisor function. In this paper, we apply the circle method to obtain an asymptotic formula for the sum

|miX|Yτk(m12+m22++m2)(i=1,2,,)
with
Y=X185(k+1)+20ε
tending to infinity. Previously, only the case k = 2 was considered.

Keywords

Higher divisor functions / almost equal variables / circle method

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Miao Lou. Sums of Higher Divisor Functions with Almost Equal Variables. Frontiers of Mathematics, 2024, 20(3): 603-615 DOI:10.1007/s11464-023-0038-5

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