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Archive for the ‘Number Theory’ Category

Why is the Riemann Hypothesis true?

“We must know;  we shall know.”  — David Hilbert.

“When the solution is simple, GOD is answering.”  — Albert Einstein.

Riemann Hypothesis states that the real part of all nontrivial zeros,

s = a ± b * i,

of the Riemann zeta function equals one-half (a = 1/2)  where 

Re(s = a ± b * i) = a,

and where the Riemann zeta function is  Σ 1/k^s  (from k = 1 to k = ∞).

Reference links:

Riemann Hypothesis;

Why is the Riemann Hypothesis true?;

Subtle relations: prime numbers, complex functions, energy levels and Riemann;

150 Years of Riemann Hypothesis;

Turing and the Computational Tradition in Pure Mathematics (


“… deepest interrelationships in analysis are of an arithmetical nature.”  — Hermann Minkowski.

In the simplest sense, the Riemann Hypothesis (RH) confirms for all positive integers,  k > 1,

there exists a prime, p, which divides k such that either

p = k


p ≤ k^( 1/2 = a = Re(s = a ± b * i) )

where s is the nontrivial zero of the Riemann zeta function.

Furthermore, only s + s’ = 1  over 0 ≤ a ≤ 1 satisfies the  divergent Euler zeta function (Harmonic Series) for which there exists infinitely prime numbers (see Euler’s proof)  where a = 1/2.  

Hence, s = 1/2 + b * i and s’ = 1/2 – b* i are the only nontrivial zeta zeros of the Riemann zeta function in the interval, 0 ≤ a ≤ 1.

Note:  s = a + b * i and s’ = a – b* i are the nontrivial zeta zeros of the Riemann zeta function in the interval, 0 ≤ a ≤ 1.

Therefore, the answer is affirmative.

The Riemann Hypothesis is true!   — David Cole.

“If you can’t explain simply, you don’t understand it well enough.” — Albert Einstein.

Reference Links:

What is the correct proof of the famous and important Polignac Conjecture?.

Romans 8:28