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

Abstract: At the simplest level, an upper bound sieve of Selberg type is a choice of $\rho(d)$, $d\leq D$, 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 $D$ compare 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.

Joint work with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

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The conference program, list of speakers, and abstracts are posted on the external website.

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