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Title:  On the real zeroes of the Hurwitz zetafunction and Bernoulli polynomials 
Authors:  Veselov, A.P. Ward, J.P. 
Issue Date:  2002 
Abstract:  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 $\frac{2n}{\pi e}$. 
Description:  This preprint has been submitted, and accepted, to the journal, Journal of Mathematical Analysis and Applications [© Elsevier]. The definitive version: VESELOV, A.P. and WARD, J.P., 2005. On the real zeroes of the Hurwitz zetafunction and Bernoulli polynomials. Journal of Mathematical Analysis and Applications, 305(2), pp. 712721, is available at: http://www.sciencedirect.com/science/journal/0022247X. 
URI:  https://dspace.lboro.ac.uk/2134/448 
Appears in Collections:  Preprints (Maths)

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