Some consequences of the Lindelöf conjecture

Abstract

Suppose that the Lindelöf conjecture is valid in the following quantitative form:

$$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$

, where ε(t) is a monotone decreasing function,$\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$. Then it is proved that for |t|≥T₀ the disk$\{ s:|s - \tfrac{1}{2} - it| \leqslant v\}$ contains at most 20v log |t| zeros of ζ(s) if$\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)}$. There exists an absolute constant A such that for |t|≥T₁ the disk$\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\}$ contains at least one zero of ζ(s). Bibliography: 2 titles.