一个除数问题的推广

On a Further Generalization of the Divisor Problem

本文推广了Ｒ．Ａ．Ｓｍｉｔｈ，Ｍ．Ｖ．Ｓｕｂｂａｒａｏ和Ｇ．Ｎｏｗａｋ所考虑的一个除数问题．令Ｓ＝｛（ａｉ，ｑｉ）｜ａｉ≤ｑｉ｝（ｒ≥３）．定义ｄ（ｎ；Ｓ）＝Σ（１）１，Σ（１）表示对满足ｎ＝ｍ１…ｍｒ，ｍｊ≡ａｊ（ｍｏｄｑｊ），ｊ＝１，２，…，ｒ的诸ｍｊ求和．我们求出了Σｎ≤ｘｄ（ｎ；Ｓ）的渐近公式，并得到了余项估计．

In this paper, we generalize a divisor problem which was considered by R. A. Smith, M. V. Subbarao and G. Nowak. Suppose r ≥3 and S = {(ai,qi)|ai≤ qi,i=1, 2,…,r}. Define d(n; S) =Σ(1) 1, where Σ(1) denotes the summation for all mi such that n = m1m2…mr, mr = aj(modqj),j = 1, 2,…, r. An asymptotic formula for Σn≤x d(n;S) and an estimation to the error term are obtained.