On the sum of a prime power and a power in short intervals

2018/07/26 by Suzuki, Yuta · 1 citation
#11L07 #11P32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.10150

Abstract

Let R_k,ℓ(N) be the representation function for the sum of the k-th power of a prime and the ℓ-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of R_1,2(N) over short intervals (X,X+H] of the length H slightly shorter than X^(1)/(2), which is shorter than the length H=X^(1)/(2)+ε in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for R_1,2(N) holds for H of the size X^0.337. Recently, Languasco and Zaccagnini (2018) extended their result to more general (k,ℓ). We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case ℓ=2 unconditionally up to some small factors.

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• Asymptotic formula for the sum of a prime and a square-full number in short intervals shorter than X^1/2

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