Optimality of the logarithmic upper-bound sieve, with explicit estimates (joint with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero)

Abstract: At the simplest level, an upper bound sieve of Selberg type is a choice of rho(d), d<=D, with rho(1)=1, such that

S = \sum_{n\leq N} \left(\sum_{d|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.

number theory

Audience: researchers in the topic

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