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