In this note the sums \(s(k,N)\) of reciprocals \(\sum \limits _{\frac {kp}{N}< x <\frac {(k+1)p}{N}}\frac {1}{x} \pmod p\) are investigated, where \(p\) is an odd prime, \(N\), \(k\) are integers, \(p\) does not divide \(N,N\geq 1\) and \(0\leq k\leq N-1\). 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 William’s congruences for the Fibonacci numbers.