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

In this note the sums s(k,N) of reciprocals [GRAPHICS] 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. (c) 2008 Mathematical Institute Slovak Academy of Sciences.