Robin's Inequality & the Riemann Hypothesis

• Robin's Inequality is a key result stating that σ(n) < e^γ n log(log n) for all n > 5040, creating an arithmetic criterion equivalent to the Riemann Hypothesis.
• The method employs effective Chebyshev estimates, Mertens’ theorem, and studies of colossally abundant numbers to achieve asymptotically sharp bounds.
• Research reveals that the normalized difference d(n) oscillates with a lim inf of zero, leading to new criteria that connect extremal integer structures with the truth of RH.

Robin's inequality is a central result in analytic number theory, explicitly connecting the behavior of the sum-of-divisors function to the Riemann Hypothesis (RH). For integers $n > 5040$, it asserts that $\sigma(n) < e^\gamma n \log\log n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma$ denotes the Euler–Mascheroni constant. Robin (1984) established that this inequality holds for all $n > 5040$ if and only if RH is true, converting an analytic hypothesis about the zeros of the Riemann zeta function into an explicitly checkable arithmetic property of integers. Recent research has focused on sharpening, generalizing, and deriving new criteria related to Robin's inequality, with particular interest in its asymptotic precision and relationships to subclasses of integers maximizing multiplicative functions.

1. Formal Statement and Equivalence to the Riemann Hypothesis

Robin's inequality is given by

$$\sigma(n) < e^\gamma n \log\log n,\qquad\forall n > 5040.$$

The normalized Robin remainder is

$$D(n) = e^\gamma n \log\log n - \sigma(n),\quad d(n) = \frac{D(n)}{n} = e^\gamma \log\log n - \frac{\sigma(n)}{n}.$$

Robin's foundational result is:

• If RH is true, then $D(n) \geq 0$ for all $n > 5040$.
• Conversely, if $D(n) < 0$ for some $n > 5040$, then RH is false.

This characterization makes violations of Robin’s inequality for $n > 5040$ both necessary and sufficient to disprove RH. As a result, research focuses on properties of $D(n)$ and $d(n)$ as $n \to \infty$ and in special integer sequences.

2. Asymptotic Tightness and Main Unconditional Result

An essential unconditional result established is: $\boxed{\liminf_{n \to \infty} d(n) = 0}$ This theorem demonstrates that the gap between the arithmetic side and the analytic bound in Robin’s inequality can get arbitrarily small along some sequence of integers. As a consequence, the inequality is asymptotically sharp: while it can never be uniformly improved for all $n$, arbitrarily small differences can be achieved on infinite subsequences even without assuming RH. The proof constructs special sequences of highly divisible integers and employs explicit estimates.

3. Proof Strategies and Fundamental Tools

The derivation relies on several analytic number theory ingredients:

• Effective Chebyshev Summatory Function Estimates: Bounds for $\vartheta(x) = \sum_{p < x} \log p$ that relate the growth of the largest prime factor in constructions of multiplicative record-holders.
• Effective Mertens’ Third Theorem (Rosser–Schoenfeld): Bounds on products involving primes, essential for controlling $\sigma(n)$ and similar multiplicative functions in terms of prime factorization. Specifically, for large $x$,

$$\prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} \sim e^\gamma \log x$$

with explicit error terms.

• Arithmetic Function Inequalities: Lemmas establishing monotonicity and comparative behavior of $d(n)$ for related values of $n$ differing in factorization structure.
• Construction of Dense Sequences: Numbers of the form $n = \prod_{p < x} p^{t-1}$, for growing $t$, are chosen to push $d(n)$ to zero as $t \to \infty$.

These ingredients show that $d(n)$ can be made arbitrarily small, but only the nonnegativity of $D(n)$ for all $n > 5040$ is equivalent to RH.

4. Colossally Abundant Numbers and New RH Criterion

The paper advances by focusing on colossally abundant (CA) numbers, which maximize $\sigma(n)/n^{1+\varepsilon}$ for some $\varepsilon>0$. For this distinguished class, much sharper criteria are obtainable:

• If RH is false: \quad $$\displaystyle\liminf_{n\to\infty,\, n\,\text{CA}} D(n) = -\infty$$
• If RH is true: \quad $$\displaystyle\lim_{n\to\infty,\, n\,\text{CA}} D(n) = +\infty$$

This dichotomy yields a new explicit criterion for RH: the thresholded growth or decay of $D(n)$ (or $\sigma(n)$) restricted to CA numbers is equivalent to the truth of RH. Such numbers serve as natural candidates for extremal behavior and potential violations, significantly narrowing the focus from all integers to a structurally meaningful subclass.

5. Oscillation, Sharpness, and Limiting Behavior

Oscillation theorems, previously studied for the Euler totient $(\varphi(n))$ and generalized to divisor sums, are crucial in proving these results. They reveal:

• Under RH, $D(n)$ for CA numbers eventually dominates any bounded function, yielding positivity.
• If RH fails, there exist $n$ such that $D(n)$ is deeply negative along CA subsequences.

The normalized difference $d(n)$ oscillates on CA numbers, with magnitude linked to the distribution of zeros of the Riemann zeta function, yet beats zero arbitrarily often unconditionally, certifying the tightness of Robin's inequality.

6. Implications for Number Theory and Future Directions

Robin's inequality is now recognized as not only an equivalence to RH but a probe of the deep interface between the additive and multiplicative structure of the integers. The unconditional result that $\liminf d(n) = 0$ sets asymptotic boundaries but does not weaken the power of Robin's criterion: only its strict positivity everywhere (for $n > 5040$) is tantamount to RH. The refined criterion for CA numbers sharpens the toolkit for both theoretical investigation and computational search, as violations (if any) must arise within this rarefied class.

Table: Robin’s Difference Behavior in Key Integer Classes

Integer Class	Behavior of $D(n)$	RH Criterion Condition
All $n > 5040$	$D(n) \geq 0$ iff RH	Robin’s classical equivalence
General $n$	$\liminf_{n\to\infty} d(n) = 0$	Asymptotic tightness (unconditional)
$n$ colossally abundant	$D(n)\to+\infty$ iff RH, else $-\infty$	New explicit CA-based criterion

Ongoing work involves exploring similar sharpness for related inequalities, extensions to other multiplicative functions, and the implementation of these analytic bounds in large-scale computational verification campaigns of Robin's inequality among abundant and CA numbers. The dependence of the new explicit RH criterion solely on limiting behavior for CA numbers underscores the continued importance of extremal integer structures in analytic number theory.