Estimates of \(\psi\), \(\theta\) for large values of \(x\) without the Riemann hypothesis (Q2792347)

The Chebyshev functions NEWLINE\[NEWLINE\theta(x)= \sum_{p\leq x}\ln p\quad\text{and}\quad\psi(x)= \sum_{\substack{ p,\alpha\\ p^\alpha\leq x}}\ln pNEWLINE\]NEWLINE are important functions in prime number theory. Various authors have investigated bounding the error term \(\max(|\theta(x)-x|,|\psi(x)-x|)\), including \textit{J. B. Rosser} and \textit{L. Schoenfeld} [Ill. J. Math. 6, 64--94 (1962; Zbl 0122.05001); Math. Comput. 29, 243--269 (1975; Zbl 0295.10036)]. Their results were improved by \textit{L. Schoenfeld} [Math. Comput. 30, 337--360 (1976; Zbl 0326.10037)], where one step in the proof involved verifying the Riemann hypothesis up to a fixed height \(A\). One object of the current paper is to show that it is unnecessary to find such a constant \(A\) and instead one can assume that there is a zero-free region of \(\zeta(s)\) of the form NEWLINE\[NEWLINE\text{Re}(s)\geq 1-{1\over R\ln|\text{Im}(s)|}\quad\text{for}\quad |\text{Im}(s)|\geq 2\pi,NEWLINE\]NEWLINE where \(R>0\) is a formal constant. Let NEWLINE\[NEWLINEX=\sqrt{{\ln x\over R}}\varepsilon_R(x)= \sqrt{{8\over\pi}} X^{1/2} e^{-X}.NEWLINE\]NEWLINE The author's main theorem states that for \(X\geq\max(8.36,{8\over R})\) NEWLINE\[NEWLINE\max(|\theta(x)-x|, |\psi(x)-x|)< x\varepsilon_R(x).NEWLINE\]NEWLINE The corollary asserts that one can take \(R= 5.69693\) when \(x\geq 3\); this result is especially useful in applications for \(x\geq\exp(10 000\)).NEWLINENEWLINE These results improve those of Schoenfeld for large enough \(x\). The proof applies the method of Rosser and Schoenfeld by dividing the critical strip \(\{s\in\mathbb{C}: 0<\text{Re}(s)< 1\}\) into four regions on each of which a sum over the zeros of \(\zeta(s)\) contained in it is defined and estimated. The paper ends with a table computed to give estimates of the error term \(|\psi(x)-x|\) for moderate values of \(x\).