Leudesdorf's theorem and Bernoulli numbers

summary:For $m\in$, $(m,6)=1$, it is proved the relations between the sums $$W(m,s)=\sum _{i=1, (i,m)=1}^{m-1} i^{-s}\,, \quad \quad s\in \,,$$ 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