Advances in Pure Mathematics
Vol.08 No.07(2018), Article ID:86317,15 pages
10.4236/apm.2018.87040
The Proof of the Generalized Piemann’s Hypothesis
S. V. Matnyak
Khmelnytsky, Ukraine
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: April 16, 2018; Accepted: July 27, 2018; Published: July 30, 2018
ABSTRACT
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.
Keywords:
Riemann’s Hypothesis, Natural Series, Function of Möbius, Mertens Function, Finite Exponential Functional Series, Finite Exponential Functional Progression
1. 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 have a real part that is equal to .
1). For this we first will provethe Riemann's hypothesis for the zeta function .
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
(1)
is called finite exponential functional series with respect to the variable exponent , where .
Definition 2. The progression of the type
(2)
is called finite exponential functional progression, if their first member is a function of or is equal to 1 and the denominat or is a function of the variable .
Theorem 1. If the set of natural numbers is the union of subsets and these subsets are disjoint and have appropriately for the elements, the number of elements of the set equal to
.
Proof. The theorem is proved similarly to the theorem 7.11 ( [2] , p. 50).
Theorem 2. of a series equals.
Proof. Let the number be the quantity of elements of the set of natural numbers . The positive integers consist: of 1; primes, the quantity of which is denoted by ; of natural numbers, which are divided on with quotient from 1 with , the number of which is denoted by ; the number of positive integers, which are decomposed into a product of a pair number of primes, we denote by , 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 , of the set of natural numbers , according to the theorem 1, equals
(3)
The number of the natural series approximately can be expressed as a finite exponential function series
(4)
We will write the number of natural numbers , approximately as finite exponential function series consisting of the first members of the series (4), and we denote it as , then
. (5)
In the sum of the series (4) each term of the series is taken, as the amount of natural numbers. For example, ; ; and so on.
A series is greater than the series of the natural number of positive integers .
Definition 3. The natural numbers that overlap, are called the finite exponential function series of (5), in which they occurmore than once. Let that and
. The function , because in the function except the numbers are included and the numbers which overlap.
Definition 4. Two infinitely great and , which are not equal to each ether are called equivalent if when .
We assume that the set of natural numbers with algebra is vector space . In this space we set the standard
, and the set of numbers with algebra will be assumed as vector space with standard
.
Then, the denominator of an exponential function of finite progression, which
operates in the space will take as , and the denominator of an exponential function of finite progression, which operates in the space , as .
The finite exponential functional series (4) is approximable by the sum of the finite exponential functional progression
(6)
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 with the finite functional progression the sum of the functional series is written in the form of
Then let us write that
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) are equivalent.
Proof. The sum of the finite exponential functional progression can be calculated by the formula
(7)
and then the limit will be equal to:
where and ( [3] , p. 67).
Therefore, in accordance with the definition 4, the functional series (4) and the finite sum of the functional progression (6) at , when ,will be equivalent.
Proposition 2 is proved.
From the expression (4) and (7) one can see that is within limit of function
.
The sum of the finite exponential functional progression (6) with equals . When . We will compare the function with the function . We find
.
Therefore, .
Lemma 1. The number of natural numbers that overlap is less than .
Proof. To prove this proposition let us denotethrough is the numbers that occur more than once in the finite exponential functional series (4) when , 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 , . Two is 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
. (8)
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
We show that
are equivalent:
.
It follows that the finite exponential functional series and the finite exponential functional progression are equivalent.
Proposition 3 is proved.
To prove the theorem, we introduce the functions series
, (9)
where And functional progression
. (10)
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
. (11)
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
Therefore, in accordance with the definition 4, the functional series (9) and a functional progression (10) are equivalent.
Proposition 4 is proved.
Proposition 5. The finite exponential functional series (9) and the finite sum of the exponential function progression (10) are equivalent when ( [3] , p. 67).
Proof. The sum of functional series is more than , 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 , let us set or , and then we obtain
.
Since then . And then we will have
, (12)
Using the definition 4 we will have
.
Therefore, a function of series (9) and the sum of functional progression (10) are equivalent.
Proposition 5 is proved.
From the expressions (10) and (12) it is clear that is within
.
Then we compare function with the function when and we obtain
.
Hence, we have that , or .
Therefore, . Let us take into account the value of the finite exponential functional series , and write that
.
Lemma 1 is proved. Then we can write that
, (13)
Using the inequality we will write that
(14)
If we substitute value of the function (14) into (13), we obtain
.
Using Lemma 1, we obtain
.
Hence; we find that
.
The value from (3) is substituted instead , we obtain
,
or
. (15)
Then we can write that appropriately of the properties of the function of Mobius- ,when ; , where is the amount of prime factors of the numbers and when n is multiple for ,
. (16)
We write that
.
Then the expression (16) takes the form
.
Therefore
. (17)
The theorem is proved.
For the Mertens function we can find a more precise estimate.
Lemma 2. The accurate assessment. in a series will be equal
.
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 because . Using this method, we define what and the expression we obtain that
.
Hence, we find that the upper limit of the value functions will be the value
,
and the lower limit is
.
Therefore, the evaluation is a more accurate estimate than when .
Lemma 2 is proved.
The theorem 2 proves that the upper limit value of the function equals
,
and the lower limit is
,
Proposition 6. when
Proof. According to the theorem 54 ( [4] , p. 114) we have that . The value is compared with , we will write that when . Hence we find that when , . Therefore, we can assume that , where is a random small number. And here we find that when .
Proposition 6 is proved.
1.2). A determination the values of coefficient .
1.2.1). Then we can write that according to the properties of Mobius function- , then ; , where k the number of prime factors of the number and , when is the multiple of for that
,
Then
.
From the expression , using the properties of Möbius function, it can be written that
.
And from the expression 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
, we find that . And here we find that .
1.2.2). Using a more precise value , we find that
.
And here we find that and from the double inequality we find that the coefficient will be in the range
1.3). Theorem 3. The series converges if and where 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 .
Proof. A necessary and sufficient condition for the validity of the Riemann’s hypothesis is the convergence of the series when ( [4] , p. 114). We find the convergence of the series, when
the series diverges.
And when we have
the series converges, where ε is an arbitrary small number.
Therefore, the series converges uniformly for , and since it is a function if , for the theorem of analytic continuation, it is also at its . Therefore, the Riemann’s hypothesis is true.
The theorem is proved.
2. Theorem 4. All Non-Trivial Zeros of Dirichle’s Function L(s, χ) Have a Real Part That Is Equal to
Proof. Let’s consider the Dirichle’s series
(18)
where is the character of modulus .
There is of such series where is the Euler’s function. Since , the series (18) converges when , as can be seen from a
comparison of this series with the series . We denote it by the sum
through series . For various characters , we obtain different functions .They are called L is the Dirichle’s functions. In studying the properties of these functions it is convenient to distinguish the cases where is the main character and when .
2.1) If than the series (18) converges in the half-plane . Let us
show from the beginning, that the partial sums are limited. We divide
the integer number from 1 to into classes of deductions by and write , . Then
.
Because of the orthogonality relations
we have
,
hence
.
Since at decreases monotonically and tends to zero when , then the series converges for real , and, consequently, for all in the half-plane when . If, however, , then this the series obviously diverge. It’s abscissa converges and the abscissa of absolute convergence . By the theorem 4, The Dirichle’s
series in the half-plane of the convergence is a regular analytic
function from , the successive derivatives of which are obtained by the term differentiation of this the series ( [6] , p. 153), the function , is a regular analytic function from when .
2.2) If we use
(19)
From the theorem 3 it follows that when . If is the main character by , then
Using the condition the function (19) can be written as , when and .
Using the results of the theorem 3, it can be argued that the generalized Riemann’s hypothesis is true, and accordingly to it: “All non-trivial zeros of the Dirichle’s functions have a real part equal to ”.
The theorem 4 is proved.
3. Appendix. Disproof of the Mertens Hypothesis
The refutation of the Mertens hypothesis can be found on the basis of the proof of the Riemann hypothesis given in this paper. Take the series
and it can be written in the form
where
.
Then we can write
or
(20)
The value from the expression 3 is substituted instead of , we obtain
From the expression (20) we obtain
. (21)
The properties of Mobius function ( [5] , p. 3) will be applied to the expression (21) and we obtain that
or
.
It will be the smallest value of the function of Mertens and the biggest value for the function of Mertens . From the expression 6
we find that
, (22)
When and ( [3] , p. 67).
We write the expression 22 in the form
. (23)
Let us apply the properties of Mobius function to the expression (23) and we obtain
.
Then we can state that the function of Mertens is within
And it rejects the hypothesis of Mertens.
4. Conclusion
In the article, based on the finite exponential functional series and the finite exponential functional progressions, we prove the generalized Riemann’s hypothesis, as well as the Riemann hypothesis. It is shown that in the Riemann’s
hypothesis . In the annex to the article, the Mertens hypothesis is refuted.
In the refuted Mertens hypothesis it is shown that the Mertens function is within .
Cite this paper
Matnyak, S.V. (2018) The Proof of the Generalized Piemann’s Hypothesis. Advances in Pure Mathematics, 8, 672-686. https://doi.org/10.4236/apm.2018.87040
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