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On the Sign of the Difference $\pi(x) - \mathrm{li}(x)$

Herman J. J. te Riele
Mathematics of Computation
Vol. 48, No. 177 (Jan., 1987), pp. 323-328
DOI: 10.2307/2007893
Stable URL: http://www.jstor.org/stable/2007893
Page Count: 6
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Abstract

Following a method of Sherman Lehman we show that between $6.62 \times 10^{370}$ and $6.69 \times 10^{370}$ there are more than $10^{180}$ successive integers $x$ for which $\pi(x) - \operatorname{li}(x) > 0$. This brings down Sherman Lehman's bound on the smallest number $x$ for which $\pi(x) - \operatorname{li}(x) > 0$, namely from $1.65 \times 10^{1165}$ to $6.69 \times 10^{370}$. Our result is based on the knowledge of the truth of the Riemann hypothesis for the complex zeros $\beta + i\gamma$ of the Riemann zeta function which satisfy $|\gamma| < 450,000$, and on the knowledge of the first 15,000 complex zeros to about 28 digits and the next 35,000 to about 14 digits.

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