Digital Sum Problems for the p-adic Expansion of Natural Numbers

抄録

Let p be a positive integer greater than 1 and denote the p-adic expansion of n ∈N by n =∑_i ≥0α_i (n )p ⁱ, where α_i (n ) ∈{0,1,…,p −1}. We set s (n,l ) =∑_i ≥01_{{αi (n)=l }} for l =1,2,…,p −1, and s (n ) =∑_l =1^p −1l s (n,l ). We define the power sum of sum of digits by S_k(N ) =∑_n =0^N −1s (n )^k, k ∈N, and the exponential sum of sum of digits by F (ξ ,N ) =∑_n =0^N −1e ^ξs(n), ξ ∈R for N ∈N. Explicit formulas of these sums have been investigated by many authors for a long time. In this paper, developing our previous results, we shall give explicit formulas of F (ξ ,N ) and S_k(N ).