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A note on some relations among special sums of reciprocals modulo $$p$$. (English) Zbl 1164.11001
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.

##### MSC:
 11A07 Congruences; primitive roots; residue systems 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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