Uniform Convergence Behavior of the Bernoulli Polynomials

Abstract

The roots of Bernoulli polynomials, $B_n(z)$, when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of $\mathbb{C}$, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of $B_n(nz)$, compute the distribution of real roots of Bernoulli polynomials and show that, properly rescaled, the complex roots lie on the curve $e^{- 2\pi \text{Im}(z)} = 2\pi e |z|$ or $e^{2\pi \text{Im}(z)}= 2\pi e |z|$.