New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities

Abstract

We obtain new sharp bounds for the Bernoulli numbers:2(2n)!/(π2n(22n−1))<|B2n|≤(2(22k−1)/22k)ζ(2k)(2n)!/(π2n(22n−1)),n=k,k+1,…, k∈N+, and establish sharpening of Papenfuss's inequalities, the refinements of Becker-Stark, and Steckin's inequalities. Finally, we show a new simple proof of Ruehr-Shafer inequality.