Frank Vega

Simple Proof for the Riemann Hypothesis

Keywords: Mathematics Subject Classification (2010). Primary 11M26; Secondary 11A41 Riemann hypothesis, prime numbers, Riemann zeta function, Chebyshev function

Let Ψ(n) = n • q|n 1 + 1 q denote the Dedekind Ψ function where q | n means the prime q divides n. Define, for n ≥ 3; the ratio

n•log log n where log is the natural logarithm. Let Mx = q≤x q be the product extending over all prime numbers q that are less than or equal to a natural number x > 1. The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part 1 2 . It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the Riemann hypothesis. In 2011, Solé and Planat stated that the Riemann hypothesis is true if and only if the inequality R(Mx) > e γ ζ(2) holds for all x ≥ 5, where γ ≈ 0.57721 is the Euler-Mascheroni constant and ζ(x) is the Riemann zeta function. In this note, using Solé and Planat criterion, we prove that the Riemann hypothesis is true.

Introduction

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2 . It was proposed by Bernhard Riemann (1859). The Riemann hypothesis belongs to the Hilbert's eighth problem on David Hilbert's list of twenty-three unsolved problems. This is one of the Clay Mathematics Institute's Millennium Prize Problems. In mathematics, the Chebyshev function θ(x) is given by θ(x) = q≤x log q with the sum extending over all prime numbers q that are less than or equal to x, where log is the natural logarithm. Proposition 1.1. For every x > 1 [6, Theorem 4 (3.15) pp. 70]:

θ(x) < 1 + 1 2 • log x • x.
The following property is based on natural logarithms:

Proposition 1.2. For x > -1 [3, pp. 1]: log(1 + x) ≤ x.
Leonhard Euler studied the following value of the Riemann zeta function (1734) [START_REF] Ayoub | Euler and the Zeta Function[END_REF].

Proposition 1.3. We define [1, (1) pp. 1070]:

ζ(2) = ∞ k=1 q 2 k q 2 k -1 = π 2 6 ,
where q k is the kth prime number. By definition, we have

ζ(2) = ∞ n=1 1 n 2 ,
where n denotes a natural number. Leonhard Euler proved in his solution to the Basel problem that

∞ n=1 1 n 2 = ∞ k=1 q 2 k q 2 k -1 = π 2 6 ,
where π ≈ 3.14159 is a well-known constant linked to several areas in mathematics such as number theory, geometry, etc.

The number γ ≈ 0.57721 is the Euler-Mascheroni constant which is defined as

γ = lim n→∞ -log n + n k=1 1 k = ∞ 1 - 1 x + 1 ⌊x⌋ dx.
Here, ⌊. . .⌋ represents the floor function. Franz Mertens discovered some important results about the constants B and H (1874) [START_REF] Mertens | Ein Beitrag zur analytischen Zahlentheorie[END_REF]. The number B ≈ 0.26149 is the Meissel-Mertens constant where γ = B + H [START_REF] Mertens | Ein Beitrag zur analytischen Zahlentheorie[END_REF].

Proposition 1.4. We have [2, Lemma 2.1 (1) pp. 359]:

∞ k=1 log q k q k -1 - 1 q k = γ -B = H.
For x ≥ 2, the function u(x) is defined as follows [5, pp. 379]:

u(x) = q>x log q q -1 - 1 q .
On the sum of the reciprocals of all prime numbers not exceeding x, we have:

Proposition 1.5. For x > 1 [6, Theorem 5 (3.17) pp. 70]:

-

1 2 • log 2 x < q≤x 1 q -B -log log x.
In number theory, Ψ(n) = n • q|n 1 + 1 q is called the Dedekind Ψ function where q | n means the prime q divides n. For x ≥ 2, a natural number M x is defined as

M x = q≤x q.
We define R(n) = Ψ(n) n•log log n for n ≥ 3. We say that Dedekind(x) holds provided that Putting all together yields a proof for the Riemann hypothesis.

R(M x ) > e γ ζ (2)

Central Lemma

The following is a key Lemma.

Lemma 2.1. The inequality q≤x ( q q-1 )

log θ(x) ≥ e γ ζ( 2
)
7
holds for all x ≥ 10 8 .

Proof. By Proposition 1.4, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ( 2
)
7
is the same as

q≤x log q q -1 -B -log log θ(x) ≥ H + 6 • γ -7 • log(ζ(2))
after of applying the logarithm to the both sides and distributing the terms. In addition,

log log θ(x) < log log 1 + 1 2 • log x • x = log log 1 + 1 2 • log x + log x = log   (log x) •   1 + log 1 + 1 2•log x log x     = log log x + log   1 + log 1 + 1 2•log x log x   ≤ log log x + log 1 + 1 2•log x log x ≤ log log x + 1 2 • log 2 x
by Propositions 1.1 and 1.2. So,

q≤x log q q -1 -B -log log x - 1 2 • log 2 x ≥ H + 6 • γ -7 • log(ζ(2)).
That is,

q≤x log q q -1 -B-log log x- 1 2 • log 2 x -u(x) ≥ H-u(x)+6•γ-7•log(ζ(2)).
after subtracting u(x) to the both sides of the inequality. By Proposition 1.4, we can see that

q≤x 1 q -B -log log x - 1 2 • log 2 x -u(x) ≥ 6 • γ -7 • log(ζ(2)).
By Proposition 1.5, we deduce that

- 1 2 • log 2 x - 1 2 • log 2 x -u(x) ≥ 6 • γ -7 • log(ζ(2)).
It is a fact that the inequality

- 1 log 2 x -u(x) ≥ 6 • γ -7 • log(ζ(2))
holds for all x ≥ 10 8 due to

- 1 log 2 x -H ≥ 6 • γ -7 • log(ζ(2)) - q≤10 8 log q q -1 - 1 q . □

Main Insight

This is the main insight.

Lemma 3.1. Dedekind(x) always holds for all x ≥ 10 8 .

Proof. By Lemma 2.1, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ (2)
7 holds for all x ≥ 10 8 . By Propositions 1.2 and 1.4, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ (2)
7 is equivalent to

e H-u(x) • R(M x ) ≥ e γ ζ (2) 7
.

Certainly, we have

q≤x q q-1 log θ(x) ≥   q≤x q q-1 e 1 q   • q≤x 1 + 1 q log θ(x) = e H-u(x) • q≤x 1 + 1 q log θ(x) = e H-u(x) • M x • q|Mx 1 + 1 q M x • log log M x = e H-u(x) • Ψ(M x ) M x • log log M x = e H-u(x) • R(M x )
using the Propositions 1.2 and 1.4 such that e 1 q ≥ 1 + 1 q for every prime q. Consequently, we would have

e H-u(x) e γ ζ(2) 6 • R(M x ) ≥ e γ ζ (2)
.

We only need to prove that e and therefore, the proof is done. □

Main Theorem

This is the main theorem.

. Proposition 1 . 6 .

 16 Dedekind(x) holds for all x ≥ 5 if and only if the Riemann hypothesis is true [7, Theorem 4.2 pp. 5].

Theorem 4 . 1 .

 41 The Riemann hypothesis is true.Proof. We already know that Dedekind(x) holds for all 5 ≤ x ≤ 10 8 [5, Theorem 3 (a) pp. 376]. In this way, the Riemann hypothesis must be true as a direct consequence of the Proposition 1.6 and Lemma 3.1. □Frank Vega NataSquad 10 rue de la Paix FR 75002 Paris France e-mail: vega.frank@gmail.com