On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials (Q1773324)

The authors investigate the real zeros of the Hurwitz zeta-function \[ \zeta(\sigma, a)=\sum_{r=0}^\infty(a+r)^{-\sigma} \] for real positive \(a\). Briefly speaking, they show that \(\zeta(\sigma, a)\) has no real zeros 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 zeros are asymptotically located at the lines \(\sigma+4a+2m=0\) with integer \(m\); if \(N(p)\) is the number of real zeros of \(\zeta(-p, a)\) with given \(p\), then \[ \lim_{p\to\infty}\frac{N(p)}{p}=\frac1{\pi e}. \] As a corollary the authors obtain the Inkeri's result that the number of real roots of the Bernoulli polynomials \(B_n(x)\) for large \(n\) is asymptotically equal to \(\frac{2n}{\pi e}\).