Stringent bounds for the non-zero Bernoulli numbers
Yogesh J. Bagul
TL;DR
The paper addresses sharpening bounds for the non-zero Bernoulli numbers $|B_{2k}|$ by leveraging Euler's product representation for the zeta function. The author derives the best possible constants $\alpha$ and $\beta$ in the bound $\frac{2\cdot (2k)!}{\pi^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\alpha)}<|B_{2k}|<\frac{2\cdot (2k)!}{\pi^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-\beta)}$, proving $\alpha=1$ and $\beta=9\left(1-\frac{8}{\pi^{2}}\right)$. The proof uses the relation $|B_{2k}|=\frac{2(2k)!}{(2\pi)^{2k}}\zeta(2k)$ and analyzes the monotone function $h(x)=3^x\left[1-\frac{1}{\left(1-2^{-x}\right)\zeta(x)}\right]$ for even integers $x\ge2$, showing $h$ decreases to $1$ and yields the optimal lower bound constant, while $h(2)$ gives the optimal upper bound constant. The results strengthen existing bounds and have potential applications in deriving precise inequalities for trigonometric and hyperbolic function expansions. The paper also proposes a conjecture for generalized prime-product bounds with explicit expressions for the optimal constants, suggesting avenues for further refinement.
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 $ α$ and $ β$ such that the double inequality $$ \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-α)} < \vert B_{2k} \vert < \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-β)}, $$ holds for $ k = 1, 2, 3, \cdots.$ Our main results refine the existing bounds of $ \vert B_{2k} \vert $ in the literature.