Optimality of the logarithmic upper-bound sieve, with explicit estimates

(joint work with Clara Aldana, Emanuel Carneiro, Carlos Andres Chirre Chavez and Julian Mejia Cordero)

At the simplest level, an upper bound sieve of Selberg type is a choice of $\rho(d),$ $d\le D$, with $\rho(1)=1$, such that $$S = \sum_{n\leq N}\left(\sum_{d\mid n}\mu(d) \rho(d)\right)^2$$

is as small as possible.

The optimal choice of $\rho(d)$ for given $D$ was found by Selberg. However, for several applications, it is better to work with functions $\rho(d)$ that are scalings of a given continuous or monotonic function $\eta$. The question is then what is the best function $\eta$, and how does $S$ for given $\eta$ and $D$ compares to $S$ for Selberg's choice.

The most common choice of $\eta$ is that of Barban-Vehov (1968), which gives an $S$ with the same main term as Selberg's S. We show that Barban and Vehov's choice is optimal among all $\eta$, not just (as we knew) when it comes to the main term, but even when it comes to the second-order term, which is negative and which we determine explicitly.