A NEW BOUND FOR THE SMALLEST x WITH π(x) > li(x)

2010/05/01 by Kuok Fai Chao, Roger Plymen · 1 citation
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Limits and Structures in Graph Theory #Mathematics #Interval (graph theory) #Font #Combinatorics #Upper and lower bounds #Arithmetic #Discrete mathematics #Mathematical analysis #Computer science

paper · doi:10.1142/s1793042110003125

openalex publication_date 2010/05/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/23

Abstract

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which π(x) - li (x) > 3.2 × 10 151 . There are at least 10 154 successive integers x in this interval for which π(x) > li (x). This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.

Citations

• The Development of Prime Number Theory
• Approximate formulas for some functions of prime numbers
• Sur la distribution des nombres premiers
• On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem
• Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes
• An Introduction to the Theory of the Riemann Zeta-Function

Cited by

• Skewes's number [wikipedia]

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