Abstract
We assume the Riemann Hypothesis (RH). It is classical that there is an absolute constant $C > 1$ such that $\psi(x)-x$ changes sign in every interval $[x, Cx]$ for $x\ge 1$. We prove that $\psi(x)-x$ changes sign in $[x,19x]$ for all $x\ge 1$, and also that for $x\ge x_0$, $\psi(x)-x$ changes sign in the interval $[x,Cx]$ where $C = 2.02$.
Citation
Hugh L. Montgomery. Ulrike M.A. Vorhauer. "Changes of sign of the error term in the prime number theorem." Funct. Approx. Comment. Math. 35 235 - 247, January 2006. https://doi.org/10.7169/facm/1229442626
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