PROOF OF RIEMANN HYPOTHESIS:

Keywords: Riemann zeta function, Power Series Expansion of the Riemann zeta Function, Fundamental Theorem of Arithmetic, Harmonic Series ([tex]H = \sum_{k=1}^{\infty }1/k = \infty[/tex]), and Complex Conjugate variables and s'.

The Riemann Hypothesis states that the real part of all non-trivial zeta zeros of the Riemann zeta function

[tex](ζ(s) = \sum_{k=1}^{\infty }1/k^{s} = 0[/tex] or [tex]ζ(s') = \sum_{k=1}^{\infty }1/k^{s'} = 0)[/tex] have a real part equal to one-half.

Fact I: The real part of all non-trivial zeros of the Riemann zeta function are located in the critical strip, the closed interval, [0, 1], according to a Riemann Theorem.

Fact II: There are infinitely many non-trivial zeros of the Riemann zeta function whose real part equals one-half according to a Hardy Theorem.

Fact III: The sum of each corresponding complex conjugate pair of non-trivial zeta zeros (s and s') of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and according to the Harmonic Series (H):

If s = a + bi, then s' = a - bi. And s + s' = 1 according to Fact III.

This equation implies 2* a = 1 or a = 1/2, and b - b = 0.

Note: Euler and others have proven that there exists an infinitely many primes in H. And the divergence of H is the key reason for that result.

Fact IV: For all k > 1, where k is a positive integer, there exists a prime number, p, so that p|k such that p = k or p ≤ [tex]\sqrt{k}[/tex] = [tex]k^{1/2}[/tex].

Therefore, a = 1/2 which is the optimal exponent value of the expression, [tex]k^{1/2}[/tex], according to Fact IV.

Thus, Riemann Hypothesis is true!

Author: David Cole

(aka primework123)