On the sign of the difference 𝜋(𝑥)-𝑙𝑖(𝑥)

te Riele, Herman J. J.

doi:10.1090/S0025-5718-1987-0866118-6

On the sign of the difference $\pi (x)-\textrm {li}(x)$
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by Herman J. J. te Riele PDF

Math. Comp. 48 (1987), 323-328 Request permission

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 $\pi (x) - {\text {li}}(x) > 0$. This brings down Sherman Lehman’s bound on the smallest number x for which $\pi (x) - {\text {li}}(x) > 0$, 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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