_{1}

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.

These are the well-known Abel’s results.

Theorem 1. Let the function

continuous and limited on every finite interval then

Corollary 1. Let the function

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

Below we will always let

For the function

Lemma 1. Let the function

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

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

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

PROOF: For

Applying the 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