The Proof of the Generalized Piemann ’ s Hypothesis

The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work [1], obtained by a finite exponential functional series and finite exponential functional progression.


Introduction
In this paper we give the proof of the generalized Riemann's hypothesison the basis of adjustments and corrections to the proof of the Riemann's hypothesis of the zeta function, which was undertaken in [1], as well as values of specified limits of the cofficient c.The paper also provides a refutation of the hypothesis of Mertens.
The formulation of the problem (Generalized Riemann's hypothesis).All non-trivial zeros of Dirichle's function ( ) L s χ have a real part that is equal to 1 2 σ = .
1).For this we first will provethe Riemann's hypothesis for the zeta function ( ) n ς .
1.1).The solution.For the confirmation of the Riemann's hypothesis we will give the definitions and prove the following theorem.
Definition 1.The expression ( ) ( ) ( ) ( ) ( ) x a x a x q x a x q x a x q x ⋅ ⋅ ⋅  is called finite exponential functional progression, if their first member ( ) a n is a function of x or is equal to 1 and the denominat or ( ) q n is a function of the variable x .

T T n =
, and the amount of numbers, which can be converted into product of unpaired number of primes, will be denoted by ( ) Then the amount of natural numbers N , of the set of natural numbers n N + , according to the theorem 1, equals The number of the natural series n N + approximately can be expressed as a finite exponential function series We will write the number of natural numbers

( )
K n , approximately as finite exponential function series consisting of the first ( ) members of the series (4), and we denote it as ( ) f n , then ( ) In the sum of the series (4) each term of the series is taken, as the amount of natural numbers.For example, We assume that the set of natural numbers ( ) In this space we set the standard max n n x n n = = , and the set of numbers , , , + ⋅ − will be assumed as vector space 2 P with standard Then, the denominator of an exponential function of finite progression, which operates in the space 1 P will take as q n = , and the denominator of an expo- nential function of finite progression, which operates in the space 2 P , as The finite exponential functional series (4) is approximable by the sum of the finite exponential functional progression Proposition 1.The finite exponential functional series (4) and the sum of the finite exponential functional progression (6) are equivalent.Proof.To prove the equivalence of the finite exponential functional series ( ) f n with the finite functional progression the sum of the functional series is written in the form of ( ) ( ) ( ) ( ) Therefore, in accordance with the definition 4, the functional series and functional progression will be equivalent.
Proposition 1 is proved.
Proposition 2. The finite exponential functional series (4) and the sum of the finite exponential functional progression (6) ( ) Proof.The sum of the finite exponential functional progression and then the limit ( ) will be equal to: where lim 1 Therefore, in accordance with the definition 4, the functional series (4) and the finite sum n → ∞ ,will be equivalent.
Proposition 2 is proved.
From the expression (4) and ( 7) one can see that The sum of the finite exponential functional progression (6) with → ∞ .We will compare the function ( ) The number of natural numbers that overlap is less than 1,5 n .
Proof.To prove this proposition let us denotethrough n K is the numbers that occur more than once in the finite exponential functional series (4) when n → ∞ , and use the exponential functional series ( ) The series ( 5) is taken is this form to be because it includes all numbers that overlap.This follows from the expression taken because it is thes mallest prime number that can not be decomposed into prime factors.
The finite exponential functional series (5) will be replace by the sum of finite exponential functional progression Proposition 3. The finite exponential functional series (5) and the finite exponential functional progression (8) are equivalent.
Proof.The functional series (4) can be written as ( ) and the functional progression ( 6) is as Discard the first members of the series and progression, we find that ( ) It follows that the finite exponential functional series ( ) f n and the finite exponential functional progression Proposition 3 is proved.
To prove the theorem, we introduce the functions series where And functional progression If we express the series ( 5), as a series , than the series (8) is taken is such form so that each element of the series (8) overlaps the each element of the series (5) with unpaired exponents of the root.And then we can write that Hence the amount of numbers that cover more numbers that overlap.
Proposition 4. The finite exponential functional series (9) and the finite exponential progression (10) are equivalent.
Proof.To prove the equivalence of the finite exponential functional series with the finite exponential functional progression in the form of the relation  Proof.The sum of functional series ( ) , and the sum of functional progression is considered, as the sum of the functional progression with Then we find that ( ) In order to calculate the functional , and then we obtain . And then we will have ( ) Using the definition 4 we will have ( ) Therefore, a function of series (9) and the sum ( ) 4 S n ⋅ of functional progression (10) are equivalent.
Proposition 5 is proved.
From the expressions (10) and ( 12) it is clear that ( ) Then we compare function n with the function ( ) f n when n → ∞ and we obtain ( ) Then we can write that appropriately of the properties of the function of Mobius-( ) We write that ( ) ( ) Then the expression (16) takes the form The theorem is proved.
For the Mertens function we can find a more precise estimate.
Lemma 2. The accurate assessment.
( ) ( ) Proof.In order to finda more accurate estimate than ( ) ( ) let us find the sum of the finite exponential functional series (5) ( ) For that we use the functional progression ( 6) , then we obtain that ( ) We find from the expression (10) that the quantity of numbers that overlap is less than ( ) . Using this method, we define what ( ) ( ) and the expression ( ) Therefore, the evaluation ( ) ( ) Lemma 2 is proved.
The theorem 2 proves that the upper limit value of the function ( ) Proof.According to the theorem 54 ( [4], p. 114) we have that ( ) .
The value ( ) ( ) , we will write that ( ) when n → ∞ .Hence we find that ( ) when n → ∞ , 0 0 ε → .Therefore, we can assume that 0 ε ε > , where ε is a random small number.And here we find that ( ) Proposition 6 is proved.

using the properties of
Möbius function, it can be written that And from the expression 0 N n − > we find that ( ) This coincides with the results [5].Then we can find the extent to which the coefficient c is located.From the double inequality ( ) . And here we find that 1.5 2 c < < .
( ) the series converges, where ε is an arbitrary small number.

Theorem 1 .
If the set of natural numbers of positive integers, which are decomposed into a product of a pair number of primes, we denote by ( ) n n find that the upper limit of the value functions

1. 2 )
. A determination the values of coefficient c .1.2.1).Then we can write that according to the properties of Mobius functionn is the multiple of m p for 2 m ≥ that

(
double inequality we find that the coefficient с will be in the range 1ε is a random small number.Corollary of Theorem 3 (the Riemann's hypothesis).All non-trivial zeros of the zeta-function have a real partequal to 1 2 σ = .Proof.A necessary and sufficient condition for the validity of the Riemann's hypothesis is the convergence of the series ( ) ).We find the convergence of the series, when