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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the sign of the difference $ \pi(x)-{\rm li}(x)$


Author: Herman J. J. te Riele
Journal: Math. Comp. 48 (1987), 323-328
MSC: Primary 11Y35; Secondary 11M06
MathSciNet review: 866118
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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) - {\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 $ \vert\gamma \vert < 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1987-0866118-6
PII: S 0025-5718(1987)0866118-6
Keywords: Prime counting function, approximation, sign changes, Riemann hypothesis, zeros of the Riemann zeta function
Article copyright: © Copyright 1987 American Mathematical Society