Alladi Ramakrishnan Hall

Parity results involving the generalized divisor function involving small prime factors of integers

Krishnaswamy Alladi

University of Florida

Let $
u_y(n)$ denote the number of distinct prime factors of $n$ that are
$denote the sum
$$
S_{-k}(x,y):=\sum_{n\leq x}(-k)^{
u_y(n)}.
$$
We describe our recent results on the asymptotic
behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, and $x$ sufficiently large. There is a crucial difference in the asymptotic behavior of $S_{-k}(x,y)$
when $k+1$ is a prime and $k+1$ is composite, and this makes the problem
particularly interesting. The results are derived utilizing a combination of
the Buchstab-de Bruijn recurrence, the Perron contour integral method, an analytic method of Selberg, and
certain difference-differential equations. The results will be described
against the background of earlier work of the first author on
sums of the M\"{o}bius function over integers with restricted prime factors
and on a multiplicative generalization of the sieve. This is joint work wih
Ankush Goswami.