Supporting compact and G-functions

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The conventions below are from Section 4.1, printed pp. 567–569 of Ramaré–Ruzsa (2001). Write $e(t)=\exp(2\pi i t)$.

Residue support and uniform lifting

Let $\mathcal K$ be a nonempty compact subset of $\widehat{\mathbb Z}$ whose images $\mathcal K_d\subseteq\mathbb Z/d\mathbb Z$ split under the Chinese remainder theorem for coprime moduli. In particular, the nonempty sets $\mathcal K_{p^\nu}$ are compatible under reduction, and $\mathcal K_d$ is the product of its prime-power factors. Set $|\mathcal K_1|=1$.

A complex sequence $(\varphi_n)_{n\geq1}$ is supported by $\mathcal K$ up to level $D$ when

$$\varphi_n\ne0\quad\Longrightarrow\quad n\bmod d\in\mathcal K_d\quad\text{for every integer }1\leq d\leq D.$$

This is equation (4.1.1). The Johnsen condition (4.1.12) additionally requires, for every prime $p$ and integer $\nu\geq1$, that

$$\#\{b\in\mathcal K_{p^{\nu+1}}:b\equiv a\pmod {p^\nu}\}$$

be independent of $a\in\mathcal K_{p^\nu}$. Compatibility of images alone is not this uniform-lifting condition.

The G-functions

For an integer $d\geq1$ and real $z>0$, equation (4.1.3) defines

$$G_d(z)=\sum_{\substack{q\leq z\\d\mid q}} \ \sum_{d\mid f\mid q}\mu(q/f)\frac{f}{|\mathcal K_f|}.$$

The equivalent nonnegative formulation (4.1.5)–(4.1.6) is

h(1)=1,\qquad h(\delta)=\prod_{p^\nu\parallel\delta} \left(\frac{p^\nu}{|\mathcal K_{p^\nu}|} -\frac{p^{\nu-1}}{|\mathcal K_{p^{\nu-1}}|}\right)\geq0, \qquad G_d(z)=\sum_{\substack{\delta\leq z\\[d,\delta]\leq z}}h(\delta),

where $[d,\delta]$ denotes the least common multiple. Thus $G_1(z)=\sum_{\delta\leq z}h(\delta)$, and $G_1(z)\geq1$ for $z\geq1$. Lemma 1 on p. 568 states that, for $\ell\mid d$,

$$G_\ell(z\ell/d)\leq G_d(z)\leq G_\ell(z).$$

These are the G-functions used in Theorem 4 and Theorem 5. They depend on the entire specified residue-support system. No asymptotic estimate for them is implied merely by this definition.

Reading. Definitions and Lemma 1 checked against printed pp. 567–569; no independent verification of the full sieve argument is asserted.