Sharper Bounds for the Chebyshev function θ(x)

2020/02/25 by Samuel Broadbent, Broadbent, Samuel, Habiba Kadiri +8 · 3 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2002.11068

openalex publication_date 2020/02/25 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this article, we provide explicit bounds for the prime counting function θ(x) in all ranges of x. The bounds for the error term for θ(x)- x are of the shape εx and (c_k x)/((log x)^k), for k=1,…,5. Tables of values for ε and c_k are provided.

Citations

• Sur la distribution des nombres premiers
• An improved upper bound for the argument of the Riemann zeta-function on the critical line II
• An explicit density estimate for Dirichlet 𝐿-series
• Explicit Bounds for Some Functions of Prime Numbers

Cited by

• Asymptotic error terms in Bonse-type inequalities
• Explicit zero-free regions for automorphic L-functions
• Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions

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