On the verification of a Nicolas inequality

2025/09/14 by Galdames-Bravo, Orlando
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary 26D07 #Secondary 11M26

paper · doi:10.48550/arxiv.2509.11182

Abstract

Nicolas inequality we deal can be written as e^γloglog N_x lt; \dfracN_xφ(N_x) , where x≥ 2, N_x denotes the product of the primes less or equal than x, γ is the Euler constant and φ is the Euler totient function. We see that verification of such an inequality depends on the sign of the big-O function in the Mertens estimate for the sum of reciprocals of primes that. Then we analyze the sign of such an error term.

Citations

• On the first sign change of θ(x) - x
• The distribution of prime numbers
• Approximate formulas for some functions of prime numbers

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• On Nicolas criterion for the Riemann Hypothesis
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• Divisor and Totient Functions Estimates
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