Riemann Hypothesis on Ramanujan's Function

EasyChair Preprint no. 9241

5 pagesDate: November 4, 2022

Abstract

Srinivasa Ramanujan studied the function $S_{1}(x) = \sum_{\rho} \frac{x^{\rho - 1}}{\rho \cdot (1 - \rho)}$ where $\rho$ runs over the nontrivial zeros of the Riemann $\zeta$ function. Under the Riemann hypothesis, we know that $\lvert S_{1}(x) \rvert \leq \frac{\tau}{\sqrt{x}}$ for $\tau = 2 + \gamma - \log(4 \cdot \pi) \approx 0.04619$. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + S_{1}(x) \cdot \sqrt{x} \cdot \log x \leq 2.062$ could be satisfied. In addition, we show that the Riemann hypothesis is true when $S_{1}(x) \geq \frac{\varepsilon}{\sqrt{x}}$ for $\varepsilon \geq -1.9999999$ and large enough $x$.

Keyphrases: Chebyshev function, Riemann hypothesis, Riemann zeta function