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

The behaviour of real zeroes of the Hurwitz zeta-function ζ(s,a) = ∞ r=0 (a +r) −s for real positive a is investigated. It is shown that ζ(s,a) has no real zeroes (s = σ, a) in the region a > −σ 2πe + 1 4πe log(−σ) + 1 for large negative σ. In the region 0 < a < −σ 2πe the zeroes are asymptotically located at the lines σ + 4a + 2m = 0 with integer m. If N(p) is the number of real zeroes of ζ(−p, a) with given p, then lim p→∞ N(p) p = 1 πe . As a corollary we have a simple proof of Inkeri’s result that the number of real roots of the classical Bernoulli polynomials Bn(x) for large n is asymptotically equal to 2n πe .  2005 Elsevier Inc. All rights reserved.