The behaviour of real zeroes of the Hurwitz zeta function $$\zeta (s,a)=\sum_{ r=0}^{ \infty}(a+r)^{ -s}\qquad\qquad a > 0$$ is investigated. It is shown that $\zeta (s,a)$ has no real zeroes $(s=\sigma,a)$ in the region $a >\frac{ -\sigma}{ 2\pi e}+\frac{ 1}{ 4\pi e}\log (-\sigma) +1$ for large negative $\sigma$. In the region $0 < a < \frac{ -\sigma}{ 2\pi e}$ the zeroes are asymptotically located at the lines $\sigma + 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $\zeta(-p,a)$ with given $p$ then $$\lim_{ p\to\infty}\frac{ N(p)}{ p}=\frac{ 1}{ \pi e}.$$ 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 $\frac{ 2n}{ \pi e}$.
Categories: math.GM math.CA