Theorem 1. Let the function be limited on every finite interval, and (x) is
continuous and limited on every finite interval then
Corollary 1. Let the function, , then
Our goal is to use this theorem on the analogs of zeta functions. We are interested in the analytical properties of the following generalizations of zeta functions:
Let N be the set of all natural numbers and ―the set of all natural numbers without
Below we will always let, this limitation is introduced only to simplify the calculations. Considering all the information above let us rewrite
For the function let us apply the results obtained by Muntz for the zeta function representation. With the help of the given definitions we formulate the analog of Muntz theorem.
Lemma 1. Let the function
PROOF: According to the theorem conditions we have
Lemma 2. Let the function
PROOF: Follows from computing of integrals.
Lemma 3. Let the function
PROOF: Computing the sums , we have
Theorem 2. Let the function
PROOF: Using Corollary 1. we have
From the last equation we obtain the regularity of the function as s satisfied
Theorem 3. The Riemann’s function has nontrivial zeros only on the line;
PROOF: For, we have
Applying the formula from the theorem 2
estimating by the module
Estimating the zeta function, potentiating, we obtain
According to the theorem 1 limited for z from the following multitude
similarly, applying the theorem 2 for we obtain its limitation in the same multitude. For the function we have a limitation for all z, belonging to the half-plane Re(s) > 1/2 + 1/R. similarly, applying the theorem 2 for we obtain its limitation in the same multitude and finally we obtain:
These estimations for prove that zate function does not have zeros on the half-plane due to the integral representation (3) these results are projected on the half-plane for the case of nontrivial zeros. The Riemann’s hypothesis is proved.
 Kawagoe, K., Wakayama, M. and Yamasaki, Y. (2008) The q-Analogues of the Riemann zeta, Dirichlet L-Functions, and a Crystal Zeta-Function. Forum Mathematicum, 20, 126.