[Paper Review] A note on the real part of the Riemann zeta-function

Juan Arias de Reyna, Richard P. Brent+1 more|arXiv (Cornell University)|Dec 21, 2011

Analytic Number Theory Research10 references3 citations

TL;DR

This paper presents an efficient algorithm to compute the critical constant $\sigma_0 \approx 1.19$, the supremum of $\sigma$ for which $\mathrm{Re\,}\zeta(\sigma + it)$ can be zero or negative for some real $t$. It uses series acceleration via the prime zeta function and inverse trigonometric series to compute $\sigma_0$ with high precision, and identifies the first 50 intervals on $\sigma = 1$ where $\mathrm{Re\,}\zeta(1+it) \leq 0$, with the first occurring at $t \approx 682112.9169$ and length $\approx 0.0529$. The sum of the first 50 interval lengths estimates the density $d(1) \approx 3.85 \times 10^{-7}$.

ABSTRACT

We consider the real part $\Re(ζ(s))$ of the Riemann zeta-function $ζ(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $σ_0 = 1.19\ldots$ which is defined to be the supremum of $σ$ such that $\Re(ζ(σ+it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re(ζ(1+it)) \le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals.

Motivation & Objective

• To compute the constant $\sigma_0 = \sup\{\sigma \in \mathbb{R} \mid \exists t \in \mathbb{R}, \mathrm{Re\,}\zeta(\sigma + it) = 0\}$ with high precision.
• To develop an efficient algorithm for evaluating $f(\sigma) = \sum_p \arcsin(p^{-\sigma}) - \pi/2$ by transforming the slowly convergent series into a rapidly convergent double series.
• To identify and tabulate the first 50 intervals on the line $\sigma = 1$ where $\mathrm{Re\,}\zeta(1+it) \leq 0$, including their locations and lengths.
• To estimate the natural density $d(1)$ of the set $\{t \in \mathbb{R}_{>0} \mid \mathrm{Re\,}\zeta(1+it) \leq 0\}$ using the sum of the first 50 interval lengths.

Proposed method

• Use the Euler product and Möbius inversion to compute the prime zeta function $P(\sigma) = \sum_p p^{-\sigma}$ via $P(\sigma) = \sum_{r=1}^\infty \frac{\mu(r)}{r} \log \zeta(r\sigma)$, which converges rapidly for $\sigma > 1$.
• Express $f(\sigma) = \sum_p \arcsin(p^{-\sigma}) - \pi/2$ as a double series $f(\sigma) = \sum_{k=0}^\infty c_k P((2k+1)\sigma) - \pi/2$, where $c_k = \frac{(2k)!}{(2^k k!)^2 (2k+1)}$, enabling faster convergence.
• Apply the inequality $c_k \leq \frac{1}{2(2k+1)}$ to bound tail errors and determine truncation points for the series in $k$ with rigorous error control.
• Use the identity $f(\sigma) = \sum_{j=1}^\infty d_j \log \zeta(j\sigma) - \pi/2$, where $d_j = \sum_{k \geq 0, r > 0, (2k+1)r = j} \frac{c_k \mu(r)}{r}$, to enable efficient evaluation using precomputed $\log \zeta(j\sigma)$ values.
• Apply a zero-finding algorithm (e.g., bisection or secant method) to locate the root of $f(\sigma)$ in an interval where $f(\sigma)$ changes sign, such as $[1.1, 1.2]$, with guaranteed error bounds.
• Apply the maximum slope principle and a bound on the logarithmic derivative $\left| \mathrm{Re} \frac{\zeta'(1+it)}{\zeta(1+it)} \right| \leq \frac{3}{4} \log(t^2 + 4) + 7$ for $t \geq 10$ to rigorously locate the first $t$ where $\mathrm{Re\,}\zeta(1+it) \leq 0$.

Experimental results

Research questions

• RQ1 What is the precise value of $\sigma_0$, the supremum of $\sigma$ such that $\mathrm{Re\,}\zeta(\sigma + it)$ can be zero or negative for some real $t$?
• RQ2 How can the slowly convergent series $\sum_p \arcsin(p^{-\sigma})$ be transformed into a rapidly convergent form for numerical evaluation?
• RQ3 What are the locations and lengths of the first 50 intervals on $\sigma = 1$ where $\mathrm{Re\,}\zeta(1+it) \leq 0$?
• RQ4 What is the estimated natural density $d(1)$ of the set $\{t > 0 \mid \mathrm{Re\,}\zeta(1+it) \leq 0\}$?
• RQ5 Is the first zero-crossing of $\mathrm{Re\,}\zeta(1+it)$ at $t \approx 682112.9169$ the global minimum, and what is the minimal value?

Key findings

• The constant $\sigma_0$ is computed to high precision using a series transformation and zero-finding algorithm, with $\sigma_0 \approx 1.19$.
• The first interval where $\mathrm{Re\,}\zeta(1+it) \leq 0$ occurs at $t \approx 682112.9169$, with a length of approximately $0.05291225$.
• The first $t$ for which $\mathrm{Re\,}\zeta(1+it) \leq 0$ is $t \approx 682112.8913$, and the local minimum value is $-0.0027652$.
• The sum of the lengths of the first 50 intervals where $\mathrm{Re\,}\zeta(1+it) \leq 0$ is $6.48390168$, yielding an estimate $d(1) \approx 3.85 \times 10^{-7}$.
• The Monte Carlo estimate of $d(1)$ is $3.80 \times 10^{-7}$, showing strong agreement with the interval sum.
• The first 50 negative local minima of $\mathrm{Re\,}\zeta(1+it)$ are listed in Table 1, with values ranging from $-0.0826$ to $-0.0008$, and interval lengths from $0.0305$ to $0.2840$.