The origin of the logarithmic integral in the prime number theorem

2013/09/30 by Kolbjørn Tunstrøm, Tunstrøm, Kolbjørn
Mathematics · #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Combinatorics #FOS: Mathematics #Function (biology) #Geometry #Logarithm #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Natural number #Number Theory (math.NT) #Physics #Pi #Prime (order theory) #Prime number #Prime number theorem #math.NT

paper · pdf · doi:10.48550/arxiv.1311.1093

31 pages; 16 figures

arxiv created 2013/09/30 · openalex publication_date 2013/09/30 · arxiv updated 2013/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We establish why li(x) outperforms x/log x as an estimate for the prime counting function pi(x). The result follows from subdividing the natural numbers into the intervals s_k :=p_k²,..., p_k+1²-1, k>=1, each being fully sieved by the k first primes p₁,..., p_k. Denoting the number of primes in s_k by pi_k, we show that pi_k |s_k|/log p_k+1² and that pi(x) li(x) originates as a continuum approximation of the sum sum_k pi_k. In contrast, pi(x) x/log x stems from sieving repeatedly in regions already completed---explaining why x/log x underestimates pi(x). The explanatory potential arising from defining s_k appears promising, evidenced in the last section where we outline further research.

Citations

• The Elementary Proof of the Prime Number Theorem: An Historical Perspective
• Sieves in Number Theory
• Mertens' Proof of Mertens' Theorem
• Are There Infinitely Many Primes?
• Chebyshev's Bias
• Elementary methods in the study of the distribution of prime numbers

Related

• Primes in the intervals between primes squared
• Algorithm, probability, and prime numbers
• Sharper bounds for the error term in the Prime Number Theorem
• A Simple Closed-Form Approximation to the Prime-Counting Function
• On |\rm Li(x)-π(x)| and primes in short intervals