Proof of the Riemann hypothesis G.R. Rezende 1
∗1
and F.M.S. Lima2
Instituto Federal de Bras´ılia, CEP 70830-450, Bras´ılia- DF, Brazil 2 Instituto de F´ısica, Universidade de Bras´ılia, P. O. Box 04455, 70919-970, Bras´ılia-DF, Brazil 29 de Junho de 2018
Resumo In the present article we prove, by means of complex variable and calculus methods, the well-known Riemann hypothesis.
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Introduction
The Riemann hypothesis is a famous conjecture made in 1859 by Bernhard Riemann in his pioneering paper on prime numbers [1]. In 1900, Riemann’s hypothesis was chosen by David Hilbert as the eighth major mathematical problem, and in 2000 he was selected as a millennium problem by the Clay Mathematics Institute. Riemann, as the title of his article indicates, wished to determine the quantity of primes in any real interval, for which he extended an observation of Euler of 1737. Euler, in his article “ Variae observationses circa infinite series ” [2], mentioned a series later called the Zeta ζ(s) :=
∞ X 1 , s n n=1
(1)
where s > 1, for the sum to converge. The function (1) can be written as an infinite product, which in modern notation is Y ζ(s) = (1 − p−s )−1 , (2) p
where p is any prime greater than or equal to two. Riemann generalized the definition of Euler to the complex numbers s = σ + it and showed that the two representations above converge whenever σ > 1. In order to extend this function ∗
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1
meromorphically to every s complex (except in s = 1, where the function has a pole with residue 1), it has implicitly proved the integral representation [3] Z ∞ s−1 x 1 dx (3) ζ(s) := 1−s (1 − 2 )Γ(s) 0 1 + ex is valid for all σ > 0. A further representation of the zeta function, valid for σ > 0, and which we will use during this article is due to Hadamard and is given by γ s e(ln(2π)−1− 2 )s Y s 1 − eρ . s ρ 2(s − 1)Γ 1 + 2 ρ
ζ(s) =
(4)
where ρ represents the nontrivial zeros of the Zeta function and γ is the EulerMascheroni constant. Riemann in his 1859 paper, [1] also defines the function ξ(s), which in modern notation [4] is expressed in the form ξ(s) :=
1 s(s − 1)π −s/2 Γ(s/2)ζ(s) 2
(5)
and which has the same zeros as the Zeta function. The function ξ(s) can be expanded in the following series of powers [4] ∞ X
2n 1 ξ(s) = a2n s − = ξ(1 − s) , 2 n=0 whose coefficients are given by Z ∞ d x3/2 ψ 0 (x) a2n = 4 dx 1
1 2
(6)
2n ln(x) x−1/4 dx , (2n!)
(7)
being ψ 0 (x) a derivative with respect to x of the function ψ(x) given below ψ(x) =
∞ X
e−n
2
πx
, ∀x > 0.
(8)
n=1
Riemann also proved, in his article, the functional relation for the Zeta function πs ζ(s) = 2s π s−1 sin Γ(1 − s)ζ(1 − s) , (9) 2 or in its symmetrical form π −s/2 Γ
s 2
ζ(s) = π (s−1)/2 Γ
1−s 2
ζ(1 − s) ,
(10)
hich extends the function for any complex plane except in s = 1. Note that the equation (9) implies that the Riemann zeta function has trivial zeros when s 2
is a negative even number due to sin( πs 2 ) cancel. In the case of positive even numbers, the product sin( πs )Γ(1 − s) does not cancel and is regular because 2 the Γ(1 − s) a simple pole. Riemann thus obtained an explicit formula, which depends on the non-trivial zeros for the quantity which he sought. Throughout the process, Riemann mentions that probably all the nontrivial zeros of the zeta function lie exactly on the critical line
