(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.

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