Proof of Riemann’s zeta-hypothesis Arne Bergstrom Paper pages 1 – 21 FAQ pages 22 – 85 1 Proof of Riemann's zeta-hypothesis By ARNE BERGSTROM Abstract Make an exponential transformation in the integral formulation of Riemann's zeta- function ζ(s) for Re(s) > 0. Separately, in addition make the substitution s −> 1 - s and then transform back to s again using the functional equation. Using residue calculus, we can in this way get two alternative, equivalent series expansions for ζ(s) of order N, both valid inside the "critical strip", i e for 0 < Re(s) < 1. Together, these two expansions embody important characteristics of the zeta-function in this range, and their detailed behavior as N tends to infinity can be used to prove Riemann's zeta-hypothesis that the nontrivial zeros of the zeta-function must all have real part ½. 1. Introduction Riemann's zeta-hypothesis from 1859 [11] is expressed as follows: CONJECTURE 1.1. The nontrivial zeros of the Riemann zeta-function ζ(s) all have real part Re(s) = ½. The Riemann zeta-hypothesis is the most famous of the few still unsolved problems on Hilbert's list of twenty-three mathematical challenges, which he presented in 1900 at the dawn of the new century [12, 18]. It is also one of the seven Millennium Problems [19] named in 2000 by the Clay Mathematics Institute. It can be shown (cf [15]) that the nontrivial zeros of the zeta-function must lie inside the "critical strip", i e for 0 < Re(s) < 1, which is the range studied in this paper. The Riemann zeta-hypothesis has been computationally verified for |Im(s)| at least up to 2.4 trillion [17]. The intriguing possibility has been suggested that the Riemann zeta-function could correspond to a quantum-physical problem with its zeros corresponding to energy eigenvalues. The underlying physical problem would then correspond to a chaotic quantum system without time-reversal symmetry [4, 5]. __________________________ Key words and phrases. Riemann’s zeta-function, exponential transformation, residues, nontrivial zeros. 2 With (σ and t are real) = s + σ i t Riemann's zeta-function ζ(s) can be defined as the following series, convergent for σ > 1, = ( ) ζ s ∑ = n 1 ∞ 1 ns This Dirichlet series can also be expressed as follows (for σ > 1), = ( ) − 1 2 ( ) −s ( ) ζ s ∑ = n 1 ∞ 1 ( ) − 2 n 1 s In Sects 2 through 5 below a modification of this latter series will be derived, giving the equivalent pair (9) and (11), which are valid also inside the critical strip. Although it will be shown that (9) and/or (11) are somewhat similar to previous results found in the literature, the approach described in the following permits a more detailed analysis, leading to a proof of Conjecture 1.1. The proof of Riemann’s zeta-hypothesis given in this paper is based on the following two fundamental properties of the Riemann zeta-function: the integral representation (1), valid for Re(s) > 0 [10, 14], = ( ) − 1 2 ( ) − 1 s ( ) Γ s ( ) ζ s d ⌠ ⌡ ⎮⎮⎮⎮⎮ 0 ∞ w ( ) − s 1 + ew 1 w

(1) the functional equation (2), valid for all s [7, 13],

( ) ζ s 2s π ( ) − s 1 ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ sin 1 2 s π ( ) Γ − 1 s ( ) ζ − 1 s

(2) 2 . Variable transformation We start by transforming the variable w in (1) as follows

w eu i e

( ) − 1 2 ( ) − 1 s ( ) Γ s ( ) ζ s d ⌠ ⌡ ⎮⎮⎮⎮⎮⎮⎮ −∞ ∞ ( ) eu s + e ( ) eu 1 u

(3) 3 The integration variable w in (1) being real, we can also set u real. Then

( ) − 1 2 ( ) − 1 s ( ) Γ s ( ) ζ s d ⌠ ⌡ ⎮⎮⎮⎮⎮⎮ −∞ ∞ e ( ) s u + e ( ) eu 1 u

(4) Consider the integrand

( ) F u e ( ) s u + e ( ) eu 1
(5) and extend u to the entire complex plane,

u + x i y Extended over the complex plane, F(u) is an analytic (meromorphic) function. 3. Poles and residues We next calculate the poles of F(u) above, i e we want to find all u that satisfy the equation

e ( ) eu 1 0 which can be verified to have the following solutions (m and n are integers, n > 1),

u + ( ) ln π ( ) − 2 n 1 i π ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ + 1 2 m The poles are thus all situated in the half-plane x > 0, and are symmetric around the real axis in conjugate pairs at half-integer values of π in the positive and negative imaginary directions. The residues of F(u) corresponding to these poles are given by the following expression

( ) Res ,n m i ( ) -1 m ( ) − 2 n 1 ( ) − s 1 π ( ) − s 1 e ( ) i ( ) + /1 2 m s π Let SN be the sum of the residues in the strip 0 < y < 2 π (i e for m = 0 and m = 1), and from n = 1 up to and including the pair of residues at x = ln((2N-1) π). Then

SN 2 ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ sin 1 2 s π e ( ) i s π π ( ) − s 1 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ ∑

n 1 N ( ) − 2 n 1 (

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