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

Abstract

The behaviour of real zeroes of the Hurwitz zeta function $$\zeta (s,a)=\sum _{r=0}^{\infty }(a+r{)}^{-s}a>0$$ 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 $$\underset{p\to \infty }{\lim }\frac{N(p)}{p}=\frac{1}{\pi e}.$$ 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 $\frac{2n}{\pi e}$.

arXiv comment9 pages, 2 figures