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

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

2006-06-15

arxiv.org/abs/math/0606361v2

Mathematics

Number Theory

5 pages, 4 figures, rearranged figures

Scientific paper

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 $y_n$:
$$y_n=\frac{n}{2\pi e}+\frac{\ln(n)}{4\pi e}+O(1)\quad\text{and}\quad
c_n=\frac{2n}{\pi e}+\frac{\ln(n)}{\pi e}+O(1).$$

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