On the sign of the difference 𝜋(𝑥)-𝑙𝑖(𝑥)
1987; American Mathematical Society; Volume: 48; Issue: 177 Linguagem: Inglês
10.1090/s0025-5718-1987-0866118-6
ISSN1088-6842
Autores Tópico(s)Advanced Algebra and Geometry
ResumoFollowing a method of Sherman Lehman we show that between 6.62 × 10 370 6.62 \times {10^{370}} and 6.69 × 10 370 6.69 \times {10^{370}} there are more than 10 180 {10^{180}} successive integers x for which π ( x ) − li ( x ) > 0 \pi (x) - {\text {li}}(x) > 0 . This brings down Sherman Lehman’s bound on the smallest number x for which π ( x ) − li ( x ) > 0 \pi (x) - {\text {li}}(x) > 0 , namely from 1.65 × 10 1165 1.65 \times {10^{1165}} to 6.69 × 10 370 6.69 \times {10^{370}} . Our result is based on the knowledge of the truth of the Riemann hypothesis for the complex zeros β + i γ \beta + i\gamma of the Riemann zeta function which satisfy | γ | > 450 , 000 |\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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