Stringent bounds for the non-zero Bernoulli numbers

Yogesh J. Bagul

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arXiv · 2303.14532

Stringent bounds for the non-zero Bernoulli numbers

Yogesh J. Bagul

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Abstract

We present new sharper lower and upper bounds for the non-zero Bernoulli numbers using Euler's formula for the Riemann zeta function. In particular, we determine the best possible constants $ \alpha $ and $ \beta $ such that the double inequality $$ \frac{2\cdot (2k)!}{\pi^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\alpha)} < \vert B_{2k} \vert < \frac{2\cdot (2k)!}{\pi^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\beta)}, $$ holds for $ k = 1, 2, 3, \cdots.$ Our main results refine the existing bounds of $ \vert B_{2k} \vert $ in the literature.

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Yogesh J. Bagul. 2023-03-25. Stringent bounds for the non-zero Bernoulli numbers. https://arxiv.org/abs/2303.14532

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