[Paper Review] On the Riemann-Hardy hypothesis for the Ramanujan zeta function

Xiao‐Jun Yang|arXiv (Cornell University)|Nov 3, 2018

Advanced Mathematical Identities15 references4 citations

TL;DR

This paper proves the Riemann-Hardy hypothesis for the Ramanujan zeta function by analyzing the Ramanujan-De Bruijn function via integral representations, series and product expansions, and Hadamard's factorization theorem. It establishes that all non-trivial zeros of the Ramanujan zeta function lie on the critical line ℜ(x) = 6, confirming Hardy's 1940 conjecture.

ABSTRACT

The Ramanujan zeta function was in $1916$ proposed by an Indian mathematician Srinivasa Ramanujan. As an analogue of the Riemann hypothesis, an English mathematician Godfrey Harold Hardy proposed in $1940$ that the real part of all complex zeros of the Ramanujan zeta function is $6$. This is the well-known Riemann-Hardy hypothesis for the Ramanujan zeta function. This article is devoted to the proof of this hypothesis derived from the Ramanujan-Rankin function. Owing to the integral representation of the Ramanujan-De Bruijn function, we establish its series. We also reduce its product using the Hadamard's factorization theorem. By a class with its series and product representations, we conclude that the real part of all zeros for Ramanujan-De Bruijn function is zero. we also obtain its products of Conrey and Ghosh and Hadamard-type for the Ramanujan-Rankin function. Based on the obtained result, we prove that the Riemann-Hardy hypothesis is true.

Motivation & Objective

• To prove the Riemann-Hardy hypothesis, which posits that all non-trivial zeros of the Ramanujan zeta function lie on the critical line ℜ(x) = 6.
• To analyze the Ramanujan-De Bruijn function X(z) as a key tool in studying the zero distribution of the Ramanujan zeta function.
• To establish the convergence and symmetry properties of the series and product representations of the Ramanujan-Rankin function ξτ(x).
• To apply Hadamard’s factorization theorem to derive the canonical product form of the Ramanujan-De Bruijn function and deduce zero location.
• To verify the functional equation ξτ(x) = ξτ(12−x) and its implications for the location of zeros via symmetry and analytic continuation.

Proposed method

• Derives the integral representation of the Ramanujan-De Bruijn function X(z) using the Mellin transform of the modular form L(ω).
• Applies the functional equation ξτ(x) = ξτ(12−x) to define the Ramanujan-Wilton function Λ(φ) = ξτ(6+iφ), which is even in φ.
• Uses the Ramanujan-Rankin function ξτ(x) = (2π)−xΓ(x)Hτ(x) to connect the zeta function to the gamma function and modular forms.
• Applies Hadamard’s factorization theorem to express ξτ(x) as an infinite product over its zeros, leading to the canonical product form.
• Transforms the zero set of X(z) into purely imaginary zeros z_m = iγ_m, implying ℜ(z_m) = 0, and maps back to x = 6 + iγ_m.
• Uses convergence theorems from Knopp’s monograph to validate the absolute convergence of the product and series representations of the function.

Experimental results

Research questions

• RQ1 Do all non-trivial zeros of the Ramanujan zeta function Hτ(x) lie on the critical line ℜ(x) = 6, as conjectured by Hardy?
• RQ2 Can the Ramanujan-De Bruijn function X(z) be represented as a convergent infinite product with zeros on the imaginary axis?
• RQ3 What is the relationship between the functional equation of ξτ(x) and the symmetry of its zeros?
• RQ4 How do the series and product representations of the Ramanujan-Rankin function support the location of its zeros?
• RQ5 Can the convergence of the product and series representations be rigorously established using classical complex analysis tools?

Key findings

• All non-trivial zeros of the Ramanujan zeta function Hτ(x) lie on the critical line ℜ(x) = 6, confirming the Riemann-Hardy hypothesis.
• The Ramanujan-De Bruijn function X(z) has purely imaginary zeros, i.e., z_m = iγ_m with γ_m > 0, implying ℜ(z_m) = 0.
• The canonical product representation of X(z) is given by X(z) = X(0) ∏_{m=1}^∞ (1 + z²/γ_m²), which is absolutely convergent.
• The Ramanujan-Wilton function Λ(φ) = X(iφ) is an even function and admits the product form Λ(φ) = Λ(0) ∏_{m=1}^∞ (1 − φ²/γ_m²).
• The zeros of ξτ(x) are located at x_m = 6 ± iγ_m, confirming that their real part is exactly 6.
• The convergence of ∑ |z_m|⁻² and ∑ |Im(z_m)|⁻² is established, which supports the validity of the infinite product factorization via Knopp’s theorems.