PreprintCC BY 4.0HAL
Inequalities involving the Riemann zeta function and implications for Bernoulli numbers
Abstract
In this short note, we prove that: If $ m \in \mathbb{N} $ and $ p_i \in \mathcal{P}_0 = \left\lbrace p_0=2, p_1 = 3, p_2 = 5, ... \right\rbrace $, the set of all positive prime numbers, then for all real numbers $ x \geq 2, $ the double inequality ∏n=0m−1(pnx/(pnx−1))× pmx/((pmx−λ)) < ζ(x) < ∏n=0m−1 (pnx/(pnx−1))× pmx/((pmx−μ)), where $ \zeta $ is the Riemann zeta function, holds with the best possible constants $ \lambda = 1 $ and $ \mu = p_m^2 \left[ 1 - \frac{6}{\pi^2} \prod_{n=0}^{m-1} \frac{p_n^2}{(p_n^2 - 1)} \right].$ This solves our recently published conjecture and gives several sharp bounds for the even-indexed Bernoulli numbers.
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BibTeX
@misc{bagul2026inequalities,
title = {{Inequalities involving the Riemann zeta function and implications for Bernoulli numbers}},
author = {Yogesh J Bagul},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05757913v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inequalities involving the Riemann zeta function and implications for Bernoulli numbers
AU - Yogesh J Bagul
PY - 2026
DA - 2026/09/21
PB - HAL
UR - https://hal.science/hal-05757913v1
AB - In this short note, we prove that: If $ m \in \mathbb{N} $ and $ p_i \in \mathcal{P}_0 = \left\lbrace p_0=2, p_1 = 3, p_2 = 5, ... \right\rbrace $, the set of all positive prime numbers, then for all real numbers $ x \geq 2, $ the double inequality ∏_n=0^m−1(p_n^x/(p_n^x−1))× p_m^x/((p_m^x−λ)) < ζ(x) < ∏_n=0^m−1 (p_n^x/(p_n^x−1))× p_m^x/((p_m^x−μ)), where $ \zeta $ is the Riemann zeta function, holds with the best possible constants $ \lambda = 1 $ and $ \mu = p_m^2 \left[ 1 - \frac{6}{\pi^2} \prod_{n=0}^{m-1} \frac{p_n^2}{(p_n^2 - 1)} \right].$ This solves our recently published conjecture and gives several sharp bounds for the even-indexed Bernoulli numbers.
ER -