Stringent bounds for the non-zero Bernoulli numbers

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.