The behaviour of real zeroes of the Hurwitz zeta function ζ(s,a)=r=0∑∞(a+r)−sa>0 is investigated. It is
shown that ζ(s,a) has no real zeroes (s=σ,a) in the region a>2πe−σ+4πe1log(−σ)+1 for large negative
σ. In the region 0<a<2πe−σ 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
p→∞limpN(p)=πe1. 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
πe2n.Comment: 9 pages, 2 figure