2020/05/19 by Robert Vojak, Vojak, Robert
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2005.09307
arxiv created 2020/05/19 · openalex publication_date 2020/05/19 · arxiv updated 2020/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Define s (n) := n- 1 σ(n) (σ(n):=∑d|nd ) and ω(n) is the number of prime divisors of n. One of the properties of s plays a central role: s (pa) > s (qb) if p < q are prime numbers, with no special condition on a, b other than a, b \geqslant 1. This result, combined with the Multiplicity Permutation theorem, will help us establish properties of the next counterexample (say c) to Robin's inequality s (n) < eγ log log n. The number c is superabundant, and ω(c) must be greater than a number close to one billion. In addition, the ratio pω(c) / log c has a lower and upper bound. At most ω(c)/14 multiplicity parameters are greater than 1. Last but not least, we apply simple methods to sharpen Robin's inequality for various categories of numbers.