Asymptotic harmonic behavior in the prime number distribution

Details Asymptotic harmonic behavior in the prime number distribution Asymptotic harmonic behavior in the prime number distribution

2011-04-19

arxiv.org/abs/1104.3617v1

Mathematics

Number Theory

Scientific paper

Euler's identity is shown to give a relation between the zeros of the Riemann-zeta function and the prime numbers in terms of $\Phi(x)=x^{-1/4}[2\sqrt{x}\Sigma e^{-p^2\pi x}\ln(p)-1]$ on $x>0$, where the sum is over all primes $p$. If $\Phi$ is bounded on $x>0$, then the Riemann hypothesis is true or there are infinitely many zeros Re $z_k>1/2$. The first 21 zeros give rise to asymptotic harmonic behavior in $\Phi$ defined by the prime numbers up to one trillion.

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