The original online version of this article (Durmagambetov , A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.

Euler Chebyshev Dirichlet Riemann Hypothesis Zeta Function Muntz Function Complex Numbers Regular Function Integral Function Representation Millennium Prize Problem
2. Results

These are the well-known Abel’s results.

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

then (8)

PROOF: According to the theorem conditions we have

Lemma 2. Let the function

then

PROOF: Follows from computing of integrals.

Lemma 3. Let the function

,

, then

PROOF: Computing the sums , we have

Theorem 2. Let the function

,

, then

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.

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