[Paper Review] On numbers satisfying Robin's inequality, properties of the next counterexample and improved specific bounds

Robert Vojak|arXiv (Cornell University)|May 19, 2020

Analytic Number Theory Research5 references21 citations

TL;DR

This paper investigates Robin's inequality, a condition equivalent to the Riemann Hypothesis, by analyzing the properties of the hypothetical smallest counterexample. It establishes that such a counterexample must be superabundant and have over 969 million distinct prime factors, with at most 1/14 of its prime powers exceeding multiplicity one. The study improves bounds on Robin's inequality for various number classes using multiplicative functions and primorial analysis.

ABSTRACT

Define $s (n) := n^{- 1} σ(n)$ ($σ(n):=\sum_{d|n}d )$ and $ω(n)$ is the number of prime divisors of $n$. One of the properties of $s$ plays a central role: $s (p^a) > s (q^b)$ 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 $s (n) < e^γ \log \log n$. The number $c$ is superabundant, and $ω(c)$ must be greater than a number close to one billion. In addition, the ratio $p_{ω(c)} / \log c$ 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.

Motivation & Objective

• To determine structural constraints on the smallest counterexample to Robin's inequality, which is equivalent to the Riemann Hypothesis.
• To establish improved, explicit upper bounds for the arithmetic functions s(n) = σ(n)/n and f(n) = n/φ(n) across specific number classes.
• To characterize the multiplicative and prime factorization properties of numbers violating Robin's inequality, particularly focusing on superabundant and Hardy-Ramanujan numbers.
• To refine Robin's inequality using the function H(n), which maps any number to a Hardy-Ramanujan number with s(n) ≤ s(H(n)) and H(n) ≤ n, enabling tighter bounds.

Proposed method

• Uses the multiplicative function s(n) = σ(n)/n and its behavior under prime power changes, proving that s(p^a) > s(q^b) whenever p < q, regardless of exponents a, b ≥ 1.
• Applies the Multiplicity Permutation Theorem to analyze how reordering prime exponents affects s(n), showing that superabundant numbers maximize s(n) for their size.
• Introduces the function M(k) = e^{e^{-γ}f(N_k)} - log N_k and defines M_k(q) = 1 + M(k)/log q to derive sufficient conditions for Robin’s inequality to hold.
• Applies the function H(n), which transforms any n into a Hardy-Ramanujan number with H(n) ≤ n and s(n) ≤ s(H(n)), enabling tighter bounds for non-Hardy-Ramanujan numbers.
• Employs primorial numbers N_k = ∏_{i≤k} p_i and analyzes s(N_k) via bounds on f(N_k) and the product ∏(1 + 1/p_i), leading to improved inequalities.
• Uses asymptotic and numerical analysis of log log n and s(n) to derive explicit bounds, including g(x) = (6/7)(e^γ log log 4x + 0.6483/log log 4x) - e^γ log log 2x < 0 for x > 209.

Experimental results

Research questions

• RQ1 What are the necessary structural properties of the smallest number c violating Robin’s inequality?
• RQ2 How many distinct prime factors must the smallest counterexample to Robin’s inequality possess?
• RQ3 What constraints exist on the multiplicities of prime powers in the smallest counterexample?
• RQ4 Can Robin’s inequality be sharpened for specific classes of numbers such as odd numbers, primorials, or non-superabundant numbers?
• RQ5 How can the function H(n) be used to systematically improve bounds on s(n) for non-Hardy-Ramanujan numbers?

Key findings

• The smallest counterexample c to Robin’s inequality must be superabundant and have ω(c) ≥ 969,672,728 distinct prime factors.
• At most ω(c)/14 of the prime power multiplicities in c can exceed 1, meaning the vast majority of its prime factors have exponent 1.
• The ratio p_{ω(c)} / log c is bounded: e^{-1/log p_{ω(c)}} < p_{ω(c)} / log c < 1, implying p_{ω(c)} < log c.
• For all i ≤ ω(c), the prime power p_i^{c_i} is bounded by min(2^{c_1 + 2}, p_i e^{M(ω(c))}), providing constraints on individual exponents.
• For odd n ≥ 17, σ(n) < (2/3)e^γ n log log(2n), improving known bounds for odd numbers.
• For all i ≥ 2 and n ≥ i, the product ∏_{k=1}^n (1 + 1/p_k) < α_i (e^γ log log N_n + 2.51 / log log N_n), with α_i = ∏_{j=1}^i (1 - 1/p_j^2), yielding sharper bounds on s(N_n).