ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 Mathematical Subject Classification: 11, 26 Okada, Tatsuya, Fukushima Medical College, Sekiguchi, Takeshi, Tôhoku Gakuin University, Shiota, Yasunobu, Tôhoku Gakuin University, and Muramoto, Katsushi, Kawaijuku Educational Institution, Japan Explicit formulas for a Generalized digital sum problem. Let p be a positive integer greater than 1 and denote the p-adic expansion of n ∈ N by n = ∑_{i≥0} α_{i}(n)p^{i}, where α_{i}(n) ∈ {0, 1, …, p − 1}. We set s(n, l) = ∑_{i≥0} 1_{{α_{i}(n)=l}} for l = 1, 2, …, p − 1, and s(n) = (s(n, 1), …, s(n, p − 1)). We introduce the following generalization of the usual power and exponential sums. For N ∈ N, k = (k_{1}, …, k_{p−1}) ∈ N^{p−1}, and ξ = (ξ_{1}, …, ξ_{p−1}) ∈ R^{p−1}, define Sk(N) = ∑_{n=0}^{N−1} ∏_{l=1}^{p−1} s(n, l)^{k_{l}} and F (ξ, N) = ∑_{n=0}^{N−1} e^{∑_{l=1}^{p−1} ξ_{l}s(n,l)}. We shall give explicit formulas of (∂^{|k|}/∂ξ_{1}^{k_{1}} ⋯ ∂ξ_{p−1}^{k_{p−1}})F (ξ, N) and Sk(N) for general p. Furthermore, we apply these results to a problem of counting the number of occurrences of a fixed block of digits.