All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple

In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta have exactly the same zeros in the critical region D := z in C: Re z in (0,1); -- All the zeros of the Riemann Zeta function located on the critical line are simple; and -- The Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line L := {z in D : Re z = 1/2}.


Introduction and Summary
The proof of the Riemann hypothesis is a problem that many mathematicians consider to be the most important problem of mathematics.Indeed, it is one of at most seven mathematics problems for which the Clay Institute has offered a million dollars for its solution.To this end, the pdf publication [1] of Bombieri presents an excellent summary-along with references, to papers and books, to (see [1]) on the spacing of prime numbers n p with increasing size [3].Physic- ists have also published results on the Riemann hypothesis: in [4], Meulens compares data about the Riemann hypothesis with solutions of two dimensional Navier Stokes equations, while others [5] have compared eigenvalues of self-adjoint operators with zeros of the Riemann Zeta function.Several papers about solutions to the Riemann hypothesis have also appeared.To this end, the papers of Violi [6], Coranson-Beaudu, [7], Garcia-Morales, [8], and Chen [9] are similar to ours, in that their proof of the Riemann hypothesis are for functions that are different from the zeta function, but which have the same zeros in D as the zeta function.
Castelvecchi, author of the article [3] makes the comment: "The Riemann hypothesis will probably remain at the top of mathematicians wish lists for many years to come.Despite its importance, no attempts so far have made much progress." We wish of course to disagree with Castelvecchi's comment at the end of the above paragraph, since we believe that we have indeed proved the Riemann hypothesis in this self-contained paper, in which we accomplish the following: 1) In §2, defining the function G and showing in detail that G has exactly the same zeros, in the critical strip, { } , including multiplicity, as the zeta function; 2) Proving the positivity of , where 3) Introducing the Schwarz reflection principle in §3, which the functions , and let ( ) . In this paper we thus derive results about the function G defined by the integral, ( ) where Γ denotes the gamma function.
The operations of Schwarz reflection, the evaluation of G ℜ and G ℑ on important intervals of R , and the operations of trapezoidal and midordinate quadrature can be readily applied to the Fourier transform representation of G, which is gotten from Equation (2) defined for 0 z ℜ > , whereas an explicit Fourier transform of ζ defined by Equation (1.3) for 1 z ℜ > does not seem to be available.

Fourier Integral Representation of G, via κ
The function ζ has many other representations, with the best known of these given by: : 1 2 , .
By setting e x y = and z it

Properties of κ, ζ and G
In this section we use the definition of G given in Equation (1.2) and the identities of Equation (2.1) to derive a functional equation for G, and to derive additional properties of κ and G.We also show in detail, that ζ and G have exactly the same zeros in D, including multiplicity, that ( ) ( ) ×R and strictly decreasing as a function of σ, for ( ] , and we determine ranges of values of and their derivatives on the real line. Let us next assign notations for the left and right half of the complex plane, the critical strip(s), and the critical line.
Definition 2.1 Let − C denote the left half of the complex plane, i.e.,

{ }
, and let { } denote the right half.Let the critical strip be defined by , and let the negative and positive critical strips D  be defined as follows:

Relevant Gamma Function Relations
We shall require the use of the following lemma: Lemma 2.2 (i.) Replacement of z with z/2 in the duplication formula for the Gamma function, to get: ) and ( ) are bounded by 1 2  π for all x ∈ R , by Equations (6.1.30)and (6.1.31) of [11]; and (iii.)That the function ( ) is an entire function [11]; Γ is analytic in C except for simple poles at z n = − ( 0,1, 2, n =  ).Proof.Item (i.) is just Equation (6.1.18) of [11] with z replaced by z/2; Items (ii.) follow from Equations (6.1.30)and (6.1.31) of [11]; and Item (iii.) is just a restatement of a result found in Chapter 16. of [11].

Bounds on κ
The next lemma describes some asymptotic bounds on the function κ, which are obtained by inspection of Equation (2.2).Lemma 2.3 For any ( ) ( ) and for x real, we have x Hence the integral ( ) ( ) is finite for any polynomial Q. Proof.The bounds of κ given in (2.4) follow by inspection of the function κ as defined in (2.2).  (c.)If f is of exact multiplicity m at 0 z , then 0 z is said to be a zero (resp., a pole) of f of multiplicity m if

Analyticity Definition of Multiplicity
z is said to be a simple zero (resp., a simple pole) of f.

Functional Equations for ζ and G
An important identity of the Riemann zeta function is the well known functional equations for ζ: This functional equation for the Riemann zeta function has many important uses, including, e.g., the analytic continuation of the zeta function to all of C .The function G also possesses a functional equation which is given in Lemma 2.5 below, which plays a similar role as the functional equation for ζ. the functional equation for G is gotten by substituting the right-hand-side of the third equation of (2.1) into (2.5), and by use of Lemma 2.2: Lemma 2.5 Let z ∈C , and let G be defined as in (2.2).Then, a functional equation for the function G, valid for all z ∈C is: This equation can also be written in the form: where K is given by ( ) ( ) ( ) and where K is non-vanishing in D.
Proof.That K is non-vanishing on D follows from Lemma 2.2.

Zeros of G and ζ in D
We prove here the G and ζ have the same zeros with the same multiplicity in D and that these zeros are isolated.non-vanishing in D, so that 0 z is also a zero of ζ of multiplicity 1 k ≥ .In addition, by taking the n th derivative of ( ) * , with non-negative integer, n, we get, and ∫ for + ∫ R , then Equation (2) yields the following definitions for ( ) 2n G and for ( ) where n denotes a non-negative integer: ) Let [ ] m G be defined for any non-negative integer m by , so that by the Cauchy-Riemann equations, , where these functions are readily shown to exist, by Lemma 2.
Lemma 2.8 Let the functions [ ] m G be defined as in Definition 2.7.Then, for all 0,1, 2, m =  , and for all ( ) is analytic on the right half plane, and hence also in D. In particular given any Proof.This result follows directly by inspection of Equation (2.2 and Lemma 2.3.We omit the straight-forward proofs.

Schwarz Reflection
We present the Schwarz reflection principle, which we define as follows: Definition 3.1 Let f be analytic in D, and real on ( ) 0, a , for some Then f can be continued analytically (i.e., reflected) across ( )

Proof of the Riemann Hypothesis
Short proofs of all of all of the results which stated in the abstract of this paper are made possible by means of two well-known methods of quadrature (see e.g., [10]), which are defined by the following lemma.  .24 The first equation of (4.2) denotes the simplest trapezoidal rule, while the second denotes the simplest midordinate rule.
Theorem 4.2 Every zero of G in the critical strip z ∈C .We derive a functional equation that relates ( ) G z and ( ) 1 G z − for all z ∈C , and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region hypothesis, i.e., that all of the zeros of G in D are located on the critical line { } connections with prime numbers, to Fermat's last theorem, and to the work of authors who have shown that the first 1.5 billion zeros of the zeta function listed with increasing imaginary parts are all simple-all of which are related to the mathematics of this subject.Similarly Wikipedia of the web [2] offers an excellent summary along with references about this subject.The magazine Nature re-F.Stenger DOI: 10.4236/apm.2023.136025403 Advances in Pure Mathematics cently published a related article about a discovery by Y. Zhang, of a conjecture 2)which is related to the well-known integral for the Riemann zeta function, de- 1), we get the Fourier integral representation of G, namely,

Definition 2. 4
Let 0 z ∈C , let m denote an integer, and let f be analytic in a neighborhood of 0 z .(a.)The function f is said to have multiplicity m at 0 z if finite c;(b.)If the multiplicity of f at 0 z is m, and if 0 c ≠ , then we shall more specifically say that f is of exact multiplicity m at 0 z ;

Lemma 2. 6 (∈ is a zero of G of multiplicity 1 k
i.) The functions G and ζ have exactly the same zeros in D, including multiplicity; and(ii.)All zeros of G in C are isolated.Proof.(i.)By inspection if the third equation of (2.1) we get, if 0 z D ≥ , then we have the identity (

of multiplicity 1 m2. 7 . 7
≥ , then by applying induction with respect to 0,1, 2, , n m =  to Equation (2.9), we conclude that the multiplicity of the zero 0 z of G is also m;(ii.)Suppose that there exists a cluster of zeros { } 1 D with a sub-sequence that has a limit point z.If z D ∈ , then G would have to vanish, by Vitali's theorem.If z is on the line { } 0 z ℜ = , then, since D + ⊂ C , and since G is analytic in + C , it follows by use of the functional equation of G, that ( ) ( ) 0 1 G z G z = = − where the point 1 z − is now located on the line { } : 1 z z ∈ ℜ = C , i.e., we are back to the previous case of the convergence of such F. Stenger DOI: 10.4236/apm.2023.136025407 Advances in Pure Mathematics a sub-sequence to a point on the interior of the right half plane, where ( ) G z is analytic and bounded, so that ( ) G z would again have to vanish identically in C . Definitions of κ  , ( ) Let G and κ be defined as in Equation (2.2), and let us define

[
end we have, by use of Taylor series exway of proceeding from the first to the second line of Equation (2.13) we used the following relations, which are valid for all ( ) right hand side of Equation (13) then shows that (

Lemma 4 . 1
Let f be a real-valued function that is continuous on a finite interval [ ] , a b of R , and twice differentiable in ( ) , then by Lemma 2.10, and by Equation (2.8), .1) Remark 3.2 The Schwarz reflection principle enables analytic continuation from D  to all of D. For example, if n denotes a non-negative integer, so that the functions ( ) ( ) ( ) 2 :