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
.