Summing : a faster elementary algorithm.

We present a new elementary algorithm that takes (measured bitwise) for computing where is the Möbius function. This is the first improvement in the exponent of for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of .

Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds.

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines square-rootable Salem numbers of degree 4 which are less than or equal to .