The asymptotics for the number of real roots of the Bernoulli polynomials

Abstract

A new short clear proof of the asymptotics for the number $c_n$ of real roots of the Bernoulli polynomials $B_n(x)$, as well as for the maximal root $y_n$: $$y_n=\frac{n}{2πe}+\frac{\ln(n)}{4πe}+O(1)\quad\text{and}\quad c_n=\frac{2n}{πe}+\frac{\ln(n)}{πe}+O(1).$$