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

The behaviour of real zeroes of the Hurwitz zeta function $$��(s,a)=\sum_{r=0}^{\infty}(a+r)^{-s}\qquad\qquad a > 0$$ is investigated. It is shown that $��(s,a)$ has no real zeroes $(s=��,a)$ in the region $a >\frac{-��}{2��e}+\frac{1}{4��e}\log (-��) +1$ for large negative $��$. In the region $0 < a < \frac{-��}{2��e}$ the zeroes are asymptotically located at the lines $��+ 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $��(-p,a)$ with given $p$ then $$\lim_{p\to\infty}\frac{N(p)}{p}=\frac{1}{��e}.$$ As a corollary we have a simple proof of Inkeri's result that the number of real roots of the classical Bernoulli polynomials $B_n(x)$ for large $n$ is asymptotically equal to $\frac{2n}{��e}$.
9 pages, 2 figures