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

Abstract

The behaviour of real zeroes of the Hurwitz zeta function $$ζ(s,a)=\sum_{r=0}^{\infty}(a+r)^{-s}\qquad\qquad a > 0$$ is investigated. It is shown that $ζ(s,a)$ has no real zeroes $(s=σ,a)$ in the region $a >\frac{-σ}{2πe}+\frac{1}{4πe}\log (-σ) +1$ for large negative $σ$. In the region $0 < a < \frac{-σ}{2πe}$ 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 $$\lim_{p\to\infty}\frac{N(p)}{p}=\frac{1}{πe}.$$ As a corollary we have a simple proof of Inkeri&#39;s result that the number of real roots of the classical Bernoulli polynomials $B_n(x)$ for large $n$ is asymptotically equal to $\frac{2n}{πe}$.