On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function

The behaviour of the real roots of the Bernoulli polynomials Bm(a) for large m is investigated. It is shown that if N(m) is the number of these roots then lim m→∞ N(m) m = 2 πe We also show that on the interval Im : − [ m−2 2πe ] < a < 1 + [ m−2 2πe ] , the roots of Bm(a) are close to the half-integer lattices: a = 0,±1/2,±1,±3/2,±2, ... if m is odd, and a = ±1/4,±3/4,±5/4,±7/4, ... if m is even. The proof is based on the relation between Bm(a) and the generalised Riemann zeta-function (Hurwitz zeta-function) ζ(s, a).