The exact number of real roots of the Bernoulli polynomials

We extend previous work by Inkeri, Leeming and Delange on the number of real roots of the Bernoulli polynomials. By these earlier methods the number of real roots could not be determined exactly in many cases. We introduce a new method that enables us to determine precisely the number of real roots in virtually every case, with rare exceptions of approximately one in 1.5×108.