Mathematische Annalen

Let $$\pi S(t)$$ denote the argument of the Riemann zeta-function, $$\zeta (s)$$ , at the point $$s=\frac{1}{2}+it$$ . Assuming the Riemann hypothesis, we present two proofs of the bound $$\begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned}$$ for large $$t$$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $$S_1(t) = \int _0^{t} S(u) \> \text{ d}u$$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $$S(t)$$ to the size of the functions $$S_1(t\pm h)-S_1(t)$$ when $$h\asymp 1/\log \log t$$ . The alternative approach bounds $$S(t)$$ directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function $$f(x) = \arctan \left(\frac{1}{x}\right) - \frac{x}{1 + x^2}$$ . This draws upon recent work by Carneiro and Littmann.

Abstract. While the boundary 3-manifold of a line arrangement in the complex plane depends only on the incidence correspondence of the line arrangement, the homotopy type of the complement depends on the relative positions of incidences. In this paper we describe the homotopy type of line arrangement complements in terms of an associated plumbed graph. For pseudo-real line arrangements this method provides an explicit description of the homotopy type and fundamental group of the complement in terms of its ordered incidence graph. The method also extends to a larger class of “unknotted” line arrangements.