Let
![$ q > 2$](abs/img3.png)
be a prime number and define
![$ \lambda_q := \left(
\frac{\tau}{q} \right)$](abs/img4.png)
where
![$ \tau(n)$](abs/img5.png)
is the number of divisors of
![$ n$](abs/img6.png)
and
![$ \left( \frac{\cdot}{q} \right)$](abs/img7.png)
is the Legendre symbol. When
![$ \tau(n)$](abs/img5.png)
is a quadratic residue modulo
![$ q$](abs/img8.png)
, then the convolution
![$ \left( \lambda_q \star \mathbf{1} \right) (n)$](abs/img9.png)
could be close to the
number of divisors of
![$ n$](abs/img6.png)
. The aim of this work is to compare the mean
value of the function
![$ \lambda_q \star \mathbf{1}$](abs/img10.png)
to the well known
average order of
![$ \tau$](abs/img11.png)
. A bound for short sums in the case
![$ q=5$](abs/img12.png)
is
also given, using profound results from the theory of integer points
close to certain smooth curves.
Received January 14 2017; revised versions received June 1 2017; June 26 2017.
Published in Journal of Integer Sequences, July 1 2017.