Advances in Pure Mathematics
Vol.06 No.13(2016), Article ID:73092,4 pages

Erratum to “The Riemann Hypothesis-Millennium Prize Problem” [Advances in Pure Mathematics 6 (2016) 915-920]

A. A. Durmagambetov

L. N. Gumilyov Eurasian National University

Copyright © 2016 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

Received: December 12, 2016; Accepted: December 25, 2016; Published: December 28, 2016

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.

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







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.