On the asymptotic behavior of sums ∑n≤xf(n){x/n}k

作者：Liuying Wu; Sanying Shi

来源：International Journal of Number Theory, 2020, 16(06): 1263-1274.

DOI：10.1142/S1793042120500657

• Fractional part sum
• exponential sums
• asymptotic behavior
• van der Corput’s inequality

Let f(t) be an arbitrary real-valued positive nondecreasing function, in this paper we prove that Sf,k(x) − Tf,k(x) = ?(f(x)x131/416(log x)26947/8320),Sf,k(x) − Tf,k(x) = Sf,1(x) − Tf,1(x) + ?(f(x)x227/796+?), where Sf,k(x) is the sum given in the title, Tf,k(x) =∫1xf(t){x t }kdt and k is a positive integer. This improves the result of Mercier and Nowak.

• 单位
  1; 爱瑞思研究院; 合肥工业大学