[Paper Review] The Nicolas and Robin inequalities with sums of two squares

William D. Banks, Derrick Hart+2 more|arXiv (Cornell University)|Oct 12, 2007

Analytic Number Theory Research7 references4 citations

TL;DR

This paper establishes that the Nicolas and Robin inequalities—key to testing the Riemann Hypothesis—hold for all but finitely many natural numbers expressible as a sum of two squares. By analyzing a broad class of sets defined by prime factorization constraints, the authors prove that only finitely many such numbers violate the inequalities, and explicitly identify all 347 exceptions in the sum-of-two-squares case.

ABSTRACT

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $σ(n)5040$, where $σ(n)$ is the sum of divisors function, and $γ$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\in\cS$. As a special case, we determine the finitely many numbers of the form $n=a^2+b^2$ that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality $n/ϕ(n)1$ our results for the Robin inequality follow at once.

Motivation & Objective

• To identify infinite families of natural numbers for which the Nicolas and Robin inequalities hold for all but finitely many elements.
• To analyze the behavior of the sum-of-divisors and Euler's totient functions in relation to the Riemann Hypothesis within specific arithmetic sets.
• To determine the exact finite set of numbers of the form $ n = a^2 + b^2 $ that violate the Nicolas inequality.
• To extend known criteria for the Riemann Hypothesis by removing the $ n > 5040 $ restriction in certain cases.

Proposed method

• Define a class $ \mathcal{S} = \{ n \in \mathbb{N} \mid \text{if } p \in \mathcal{Q} \text{ and } p \mid n, \text{ then } p^2 \mid n \} $, where $ \mathcal{Q} $ is the complement of a set $ \mathcal{P} \subset \mathbb{P} $ with positive lower and sub-1 upper density.
• Use effective bounds on primes in arithmetic progressions modulo 4 to analyze the sum-of-two-squares case.
• Leverage the inequality $ \sigma(n)/n < n/\varphi(n) $ for $ n > 1 $, so results on the Nicolas inequality imply results on the Robin inequality.
• Apply Mertens’ theorem and the Prime Number Theorem to analyze asymptotic behavior of $ \sigma(n)/(n \log \log n) $ and $ n/(\varphi(n) \log \log n) $.
• Construct sequences $ \{a_n\} \subset \mathcal{S} $ satisfying $ a_n = \exp(n^{1+o(1)}) $ and $ v(p,n) \to \infty $ for each prime $ p $, to prove the lim sup of the ratio approaches $ e^\gamma $.
• Use effective bounds from [6] to compute the exact finite set $ \mathcal{S} \setminus \mathcal{N} $, resulting in 347 numbers violating the Nicolas inequality.

Experimental results

Research questions

• RQ1 For which infinite subsets $ \mathcal{S} \subset \mathbb{N} $ does the Nicolas inequality $ n/\varphi(n) < e^\gamma \log \log n $ hold for all but finitely many $ n \in \mathcal{S} $?
• RQ2 How many numbers of the form $ n = a^2 + b^2 $ fail to satisfy the Nicolas inequality?
• RQ3 Can the Riemann Hypothesis be characterized by the Robin inequality without the $ n > 5040 $ restriction for certain arithmetic sets?
• RQ4 What is the asymptotic behavior of $ \sigma(n)/(n \log \log n) $ and $ n/(\varphi(n) \log \log n) $ over the set of sums of two squares?
• RQ5 Is the lim sup of $ \sigma(n)/(n \log \log n) $ over $ \mathcal{S} $ equal to $ e^\gamma $, and does this hold for the sum-of-two-squares set?

Key findings

• The set $ \mathcal{S} \setminus \mathcal{N} $, consisting of numbers not satisfying the Nicolas inequality, is finite for any set $ \mathcal{S} $ defined by the condition that primes from a set $ \mathcal{Q} $ (with positive lower and sub-1 upper density) must appear with square factors.
• For the specific case where $ \mathcal{P} $ is the set of primes $ \equiv 1 \pmod{4} $ and $ \{2\} $, the set $ \mathcal{S} $ includes all numbers expressible as a sum of two squares.
• The set $ \mathcal{S} \setminus \mathcal{N} $ contains exactly 347 natural numbers, and among them, 246 are expressible as a sum of two squares.
• The lim sup of $ \sigma(n)/(n \log \log n) $ and $ n/(\varphi(n) \log \log n) $ over the set of sums of two squares is exactly $ e^\gamma $, confirming the asymptotic extremality of these ratios.
• The results imply that the Robin inequality holds for all but finitely many $ n $ in any such set $ \mathcal{S} $, and in particular for all but finitely many $ n = a^2 + b^2 $.
• The paper establishes that the Riemann Hypothesis is equivalent to the Robin inequality holding for all $ n $ divisible by the fifth power of some prime, without requiring $ n > 5040 $.