Abstract
In order to study integers with few prime factors, the average of
$\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$
has been a central object of research. One of the more important cases,
$k=2$
, was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For
$k\geq 2$
, it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve.
Let
$\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$
, where
$\unicode[STIX]{x1D707}_{j}$
denotes the Liouville function for
$(j+1)$
-free integers, and
$0$
otherwise. In this paper we evaluate the average value of
$\unicode[STIX]{x1D6EC}_{j,k}$
in a residue class
$n\equiv a\text{ mod }q$
,
$(a,q)=1$
, uniformly on
$q$
. When
$j\geq 2$
, we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for
$\unicode[STIX]{x1D6EC}_{k}(n)$
involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported
${\mathcal{C}}^{2}$
function.
Publisher
Cambridge University Press (CUP)