Recent investigations into number theory, particularly concerning inequalities related to arithmetic functions and the distribution of prime numbers, offer fresh perspectives on the Riemann Hypothesis. This fundamental conjecture in mathematics concerns the nontrivial zeros of the Riemann zeta function, stating they all lie on the critical line with real part 1/2. While the connection may not be immediately obvious, inequalities involving functions like the sum of divisors $\sigma(n)$ and products over primes can provide crucial insights.
One promising avenue involves the study of inequalities tied to the divisor function $\sigma(n)$. A known result establishes an equivalence between the Riemann Hypothesis and the inequality $\sigma(n) < e^{\gamma} n \log \log n$ for all integers $n > 5040$, often referred to as Robin's inequality. Research in this area focuses on:
Another framework involves inequalities related to products over prime numbers. Expressions like $\prod_{i=1}^{m} \frac{q_{i}}{q_{i}-1}$ or $\prod_{i=1}^{m} \frac{q_{i}+1}{q_{i}}$, where $q_i$ are primes, appear in attempts to bound arithmetic functions. These products are intimately linked to the Euler product representation of the zeta function:
$ \zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} $
Understanding the behavior of truncated or modified prime products can potentially reveal information about the analytic continuation and zero-free regions of the zeta function.
Furthermore, inequalities involving logarithmic terms and functions like $\operatorname{core}(n)$ (the square-free kernel of $n$) suggest connections between the distribution of prime factors and criteria related to the Riemann Hypothesis. Exploring inequalities such as $\frac{\pi^{2}}{6} \times \log \log \operatorname{core}(n) \leq \log \log n$ provides a bridge between the multiplicative structure of integers and properties relevant to the conjecture.
Combining these frameworks suggests novel research directions:
1. Refined Robin's Inequality: Instead of the strict Robin's inequality, investigate a refined version $\sigma(n) < e^{\gamma} n \log \log n + R(n)$, where $R(n)$ is a carefully chosen remainder term that depends on the prime factorization of $n$. The goal is to find an $R(n)$ such that this refined inequality is equivalent to the Riemann Hypothesis. This requires analyzing the numbers that come closest to violating the original inequality and understanding their prime structure.
2. Truncated Euler Product Analysis: Study the properties of truncated Euler products at $s=1$ and their relationship to $\log n + \gamma$. Precise bounds on the difference between the full (divergent) product and its truncations could provide insights into the distribution of primes, which in turn is linked to the zeta function's zeros via the Prime Number Theorem.
The study of these inequalities also reveals potential tangential connections to other classical number theory conjectures, such as Goldbach's Conjecture or the Erdos-Suranyi Conjecture. While not directly equivalent, formal mathematical bridges can sometimes be drawn between the techniques used to study prime distributions in these conjectures and those relevant to the Riemann Hypothesis.
A detailed research agenda based on these ideas would include:
For instance, one could analyze the refined inequality for simple cases like prime numbers or powers of primes to understand how $R(n)$ might behave. The research outlined here, rooted in the analysis of arithmetic inequalities, offers promising new angles for tackling the enduring challenge of the Riemann Hypothesis, building on work such as that presented in arXiv:03227797.