A note on some relations among special sums of reciprocals modulop

✍ Scribed by Ladislav Skula

• Book ID: 111492932
• Publisher: SP Versita
• Year: 2008
• Tongue: English
• Weight: 141 KB
• Volume: 58
• Category: Article
• ISSN: 0139-9918
• DOI: 10.2478/s12175-007-0050-3

✦ Synopsis

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.