A smooth version of Landaus explicit formula

2023/11/07 by Eugenio P. Balanzario, Balanzario, Eugenio P., Daniel Eduardo Cardenas Romero +3
Mathematics · #11N37 (Primary) #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2311.04347

openalex publication_date 2023/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a smooth version of Landaus explicit formula for the von Mangoldt arithmetical function. Assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number is a prime number, it is sufficient to know the location of a number of non trivial zeros of the Riemann zeta function. Next we use Heisenbergs inequality to support the conjecture that this number of zeros cannot be essentially diminished

Related

• Primes in arithmetic progressions under the presence of Landau-Siegel zeroes
• The Mangoldt function and the non-trivial zeros of the Riemann zeta function
• Statistical regularities in the zeta zeros
• The Landau function and the Riemann Hypothesis
• The Landau's problems. I-II