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 $$\frac{2\cdot (2k)!}{{\pi }^{2k}(2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\alpha )}<|B_{2k}|<\frac{2\cdot (2k)!}{{\pi }^{2k}(2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\beta )},$$ holds for $k=1,2,3,\cdots .$ Our main results refine the existing bounds of $|B_{2k}|$ in the literature.

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