Estimates of Some Functions Over Primes without R.H.

2010/02/02 by Pierre Dusart, Dusart, Pierre · 16 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1002.0442

openalex publication_date 2010/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.

Cited by

• On weird numbers with high abundancy index
• A Note on Algorithms for Computing p_n
• Resolution of two conjectures by Erdős and Hall concerning separable numbers
• A Dynamical Criterion Equivalent to the Riemann Hypothesis
• Small Prime Primitive Roots in Arithmetic Progressions
• Alpay Folded Prime Enumerator via gcd and Floors: Exact Enumeration, Record-Lift, and Non-Synonymy/Minimality Certificates
• On the constant factor in several related asymptotic estimates
• Finding Large Primes
• Cannonball Polygons with Multiplicities
• Inequalities For The Primes Counting Function
• Behaviour of the sequence ϑ_n = ϑ(p_n)
• An effective analytic recurrence for prime numbers
• On the arithmetic and geometric means of the prime numbers
• Derived Ramanujan Primes : R'_n
• New bounds for the sum of the first n prime numbers
• Squarefree Integers And Extreme Values Of Some Arithmetic Functions

Related

• A Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers
• Analytic implications from the remainder term of the prime number theorem
• On the prime zeta function and the Riemann hypothesis
• Experiments with zeta zeros and Perron's formula
• On the connection between zero-free regions and the error term in the Prime Number Theorem