Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers

Pub Date : 2024-07-17 DOI:10.1016/j.jnt.2024.06.012
Yuuya Yoshida
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Abstract

For all α1,α2(1,2) with 1/α1+1/α2>5/3, we show that the number of pairs (n1,n2) of positive integers with N=n1α1+n2α2 is equal to Γ(1+1/α1)Γ(1+1/α2)Γ(1/α1+1/α2)1N1/α1+1/α21+o(N1/α1+1/α21) as N, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when α1,α2(1,2) lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.

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涉及 Piatetski-Shapiro 数的二进制加法问题的渐近和非渐近结果
对于所有与 ,我们证明与 的正整数对的个数等于 ,其中 Γ 表示伽马函数。此外,我们还展示了同一计数问题的非渐近结果,即当位于比上述更大的范围时。最后,我们以启发式方法给出了类似计数问题的一些渐近公式。
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