Complex Zeros of Bernoulli Polynomials

Abstract

Bernoulli numbers and Bernoulli polynomials are named after the Swiss mathematician Jacob Bernoulli (1654-1705). He introduced these numbers and polynomials in his book Ars Conjectandi, published posthumously (Basel, 1713). These are the Bernoulli polynomials Bn(x) defined by zexz e z − 1 = X∞ n=0 Bn(x) z n n! . where the variable x and z can be real or complex and the rational numbers Bn = Bn(0) are called Bernoulli numbers. The roots of Bernoulli polynomials, Bn(x), when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of C, the Bernoulli polynomials asymptotically behave like sine or cosine. Here the asmptotic behavior of Bn(nx) discussed, compute the distribution of real roots of Bernoulli polynomials and show that properly rescaled the complex roots lie on the curve. e −2πIm(z) = 2π |z| or e2πIm(z) = 2πe |z| . Adria Ocneanu [12] showed his geometry class pictures suggesting the roots of Bernoulli polynomials lie on a distinct curve. To prove this result, the author learned about the asymptotic behavior of the Bernoulli polynomials. Gabor Szeg¨o showed in 1928, the roots of Pn(z) = Pn k=0 z k k! lie on the curve e Re(z) = e |z| . Jean Dieudonne read Szeg¨o’s paper and prove it a different way in 1935. In this paper the complex roots of Bernoulli polynomials cluster around the axis were discussed and showed by the graph.