# Properties of Prime Numbers and Nontrivial Zeta Zeros

### Properties of Prime Numbers and Nontrivial Zeta Zeros

Please consider the following important statement (highlighted blue...Go Blue! ):

The nth prime, $$p_{n}$$, exists if and only if the nth nontrivial zeta zero, $$z_{n }$$, of the Riemann Zeta Function exists.

Therefore, each prime number is unique, and so is its corresponding simple nontrivial zeta zero. The nontrivial zeta zeros are simple since their corresponding primes are also simple or unique. And primes span all the integers most efficiently according to the Prime Number Theorem.

Moreover, every positive integer, k >1, there exists a prime number, p, which divides k such that either p = k or p $$\le k^{1/2}$$. And therefore,

and that exponent of k which is the optimum 1/2 confirms the truth of the Riemann Hypothesis, Re($$z_{n }$$) = 1/2.

Of course, there are 'authorities/experts' or nonbelievers who cannot accept the truth of Riemann Hypothesis for whatever fictional or personal reasons. And that's a sad truth for some unfortunately. Truth of the Riemann Hypothesis is much more than a sound assumption -- It's the truth! Thank Lord God! Amen! Guest

### Re: Properties of Prime Numbers and Nontrivial Zeta Zeros

Guest wrote:Please consider the following important statement (highlighted blue...Go Blue! ):

The nth prime, $$p_{n}$$, exists if and only if the nth nontrivial zeta zero, $$z_{n }$$, of the Riemann Zeta Function exists.

Therefore, each prime number is unique, and so is its corresponding simple nontrivial zeta zero. The nontrivial zeta zeros are simple since their corresponding primes are also simple or unique. And primes span all the integers most efficiently according to the Prime Number Theorem.

Moreover, for every positive integer, k >1, there exists a prime number, p, which divides k such that
either p = k or p $$\le k^{1/2}$$.

And therefore, that exponent of k which is the optimum 1/2 confirms the truth of the Riemann Hypothesis, Re($$z_{n }$$) = 1/2.

Of course, there are some 'authorities/experts' or nonbelievers who cannot accept the truth of Riemann Hypothesis for whatever fictional or personal reasons. And that's a sad truth for some unfortunately. Truth of the Riemann Hypothesis is much more than a sound assumption -- It's the truth! Thank Lord God! Amen! Guest

### Re: Properties of Prime Numbers and Nontrivial Zeta Zeros

On a Proof of the Riemann Hypothesis (RH) ...

Please refer to the following link for details of an extensive proof of RH.

https://www.math10.com/forum/viewtopic.php?f=63&t=1549
Guest

### Re: Properties of Prime Numbers and Nontrivial Zeta Zeros

RH is for Right Harmony. And therefore, H (Re($$z_n$$) = 1/2) is that Harmony.