E-mail: nick1@luckynet.co.il
Abstract. For $m\in \N$, $(m,6)=1$, it is proved the relations between the sums $$ W(m,s)=\sum_{i=1, (i,m)=1}^{m-1} i^{-s}\,, \quad \quad s\in \N\,, $$ and Bernoulli numbers. The result supplements the known theorems of C. Leudesdorf, N. Rama Rao and others. As the application it is obtained some connections between the sums $W(m,s)$ and Agoh's functions, Wilson quotients, the indices irregularity of Bernoulli numbers.
AMSclassification. Primary 11A07; Secondary 11B68
Keywords. Wolstenholme-Leudesdorf theorem, $p$-integer number, Bernoulli number, Wilson quotient, irregular prime number