_{1}

This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied; these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.

In this work we are studying the properties of modified zeta functions. Riemann’s zeta function is defined by the Dirichlet’s distribution

absolutely and uniformly converging in any finite region of the complex z-plane, for which

where p is all prime numbers.

where

This equation is called the Riemann’s functional equation.

The Riemann’s zeta function is the most important subject of study and has a plenty of interesting generalizations. The role of zeta functions in the Number Theory is very significant, and is connected to various fundamental functions in the Number Theory as Mobius function, Liouville function, the function of quantity of number divisors, and the function of quantity of prime number divisors. The detailed theory of zeta functions is showed in [

The most significant contribution to the study of zeta functions is found in the results obtained by Muntz [

Muntz generalized all the results from the studies of zeta functions’ analytical properties. He noticed that all the properties can be integrated in one theory, which is called the Muntz theorem for zeta functions.

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:

where p are prime numbers. The forms of the given function (5)-(8) allow assuming that they possess the same properties as the zeta function (1), but it is not quite obvious, considering

we see the necessity of analyzing (5)-(8) functions for a deeper understanding of the properties of zeta functions.

These are the well-known results obtained by Muntz for the zeta function.

Theorem 1. Let the function

Let N be the set of all natural numbers and

Below we will always let m > 3, this limitation is introduced only to simplify the calculations. Considering all the information above let us rewrite

For the function

Theorem 2. Let the function F(x) be limited on every finite interval and have an order

PROOF: According to the theorem conditions we have

After the substitution of variables nx = y we can rewrite

The last steps are true and result from the theorem conditions and Weierstrass theorem of uniform convergence of improper integrals. Let us introduce the functions

According to the theorem conditions we have

Applying the theorem conditions we have

Substituting the variablles of the last part

Calculating we obtain the following

According to the result above we obtain

Using the properties of defined integrals and subintegral function positivity, we have

From the result above it follows that

According to the Muntz theorem, we have

Finally, after the substitution of variables we have

From the last equation we obtain the Muntz formula. From which we have the regularity of the function

Theorem 3. The Riemann’s function has nontrivial zeros only on the line

PROOF: For

Applying the Muntz formula from the theorem 2

estimating by the module

Estimating the zeta function, potentiating, we obtain

According to the theorem 1

similarly, applying the theorem 2 for

These estimations for

In this work we obtained the estimation of the Riemann’s zeta function logarithm outside of the line

The author thanks S.N. Baibekov for introducing the prime numbers to the proble- matics in the collective article [

Durmagambetov, A.A. (2016) The Riemann Hypothesis-Mil- lennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. http://dx.doi.org/10.4236/apm.2016.612069