A note on some relations among special sums of reciprocals modulo p

Abstract

Abstract In this note the sums s(k, N) of reciprocals $$\sum\limits_{\tfrac{{kp}}{N} < x < \tfrac{{(k + 1)p}}{N}} {\tfrac{1}{x}(mod p)} $$ are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers.