Counting primes by sums of frequencies

2016/07/19 by Alejandro Miralles, Miralles, Alejandro, Damià Torres +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11A41 #Secondary 11B37

paper · pdf · doi:10.48550/arxiv.1607.05517

openalex publication_date 2016/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the sequence (a_n) ⊂ (0,1] and prove that the asymptotic behaviour of ∑_k=1ⁿ a_k is the same than π(n), the prime-counting function. We also obtain that π(n) ∼ n a_n and we estimate (1)/(a_n)-(n)/(π(n)) showing that lim_{n → ∞} (1)/(a_n)-(n)/(π(n)) is convergent.

Related

• On a sum involving the prime counting function π(x)
• On a recursively defined sequence involving the prime counting function
• Asymptotics of the number of primes and sums of functions of primes in a subset of natural series
• Improving the Estimates for a Sequence Involving Prime Numbers
• On the Sum of Consecutive Primes