A Simple Closed-Form Approximation to the Prime-Counting Function

2026/02/23 by Kimihiro Noguchi
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Markov Chains and Monte Carlo Methods #Experimental and Theoretical Physics Studies

paper · doi:10.1080/00029890.2026.2621649

Abstract

Let π(x) be the prime-counting function, which counts the number of prime numbers less than or equal to some x∈R. If we let A(x)= log x−x/π(x), A(x)/ log x∼1/ log x∼π(x)/x as x→∞ [1]. In fact, the ...

Related

• New bounds for the prime counting function π(x)
• An algorithm for the prime-counting function of primes larger than three
• The prime counting function
• Prime Counting Function Identity
• Computing π(x) Analytically