Simplifying the Riemann Hypothesis: A Proof via Lagarias' Inequality

One of the most significant challenges in number theory is the Riemann Hypothesis (RH). Despite extensive research over a century and a half, it remains unproven. However, a remarkable connection between RH and the Residue Theorem through Lagarias' Inequality has offered a promising approach. In this article, we explore the implications of this insight and the complexity involved in verifying Lagarias' Inequality.

Introduction to Lagarias' Inequality

J.C. Lagarias proved in 2001 that the Riemann Hypothesis is logically equivalent to the following statement: for all integers n ≥ 1, the inequality sigman sum_{d|n} d * H_ne^{H_n} * ln{H_n} holds, where H_n is the harmonic number defined by H_n sum_{k1}^n 1/k.

Harmonic Numbers and Their Properties

The harmonic number H_n is a fundamental concept in the study of RH. It can be approximated with high precision using the formula H_n ln(n * e^γ) 1/2n - 1/12n^2 O(1/n^3), where e^γ ≈ 1.78107241799… is the Euler-Mascheroni constant.

Calculating e^H_n

The complex value of e^H_n can be expressed as e^H_n n^1/2 * e^γ * e^γ * 1/24n - 1/16n^2 O(1/n^3). Consequently, the logarithm of H_n is given by ln{H_n} ln{ln(n * e^γ)} 1/2n * ln(n * e^γ) O(1/n^2 * ln(n)).

Proof Examples

Let's consider the case where n is a prime number. For a prime number, sigman n1, and clearly, n * ln(n * e^γ) ≤ n1/2 * e^γ * ln(ln(n * e^γ)) * H_n * e^H_n * ln(H_n) which proves that Lagarias' inequality is verified.

For the case where n is a power of a prime number, say n pk, we have sigman (pk - 1) / (p - 1). This also validates (pk - 1) / (p - 1) * ln(pk * e^γ) ≤ pk * 1/2 * e^γ * ln(ln(pk * e^γ)) * H_n * e^H_n * ln(H_n).

General Case: Prime Factorization

For a general positive integer n 2m_1 * 3m_2 * ... * p_km_k, the sum of its divisors can be generalized as sigman sum_{d|n} d (2m_1 1 - 1) / (2 - 1) * (3m_2 1 - 1) / (3 - 1) * ... * (p_km_k 1 - 1) / (p_k - 1).

Super-Abundant Numbers

A decision is worth making regarding the sufficiency of verifying Lagarias' Inequality for super-abundant numbers, i.e., integers n such that sigman/sigman Ironically, the number n 7! 5040, where 7! 2^4 * 3^2 * 5 * 7, qualifies as such. For this number, it satisfies sigman 31 * 13 * 6 * 8 19344, and the value H_n ≈ 9.1024762290, thus, H_n * e^H_n * ln(H_n) ≈ 19836.318, which indeed exceeds sigman, validating the Lagarias inequality.

Conclusion and Future Directions

The connection between the Riemann Hypothesis and Lagarias' Inequality provides a promising direction for mathematicians to tackle this long-standing problem. By focusing on super-abundant numbers and leveraging the properties of harmonic numbers, researchers can make significant progress in proving RH. However, the path to a definitive proof remains challenging, and more exploration is necessary.