  Basing on properties of the Mellin transform and Ramanujan's identities,
which represent a ratio of products of Riemann's zeta- functions of different
arguments in terms of the Dirichlet series of arithmetic functions, we obtain a
number of the Poisson, M\"{u}ntz, M\"{o}bius and Voronoi type summation
formulas. The corresponding analogs of the M\"{u}ntz operators are
investigated. Interesting and curious particular cases of summation formulas
involving arithmetic functions are exhibited. Necessary and sufficient
conditions for the validity of the Riemann hypothesis are derived.
