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Asymptotic harmonic behavior in the prime number distribution

By Maurice H. P. M. van Putten


We consider $\Phi(x)=x^{-\frac{1}{4}}\left[1-2\sqrt{x}\Sigma e^{-p^2\pi x}\ln p\right]$ 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~$z_k>\frac{1}{2}$. The first 21 zeros give rise to asymptotic harmonic behavior in $\Phi(x)$ defined by the prime numbers up to one trillion.Comment: minor revision, 13 pages, 3 figure

Topics: Mathematics - Number Theory
Year: 2014
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