Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.1

Long and Short Sums of a Twisted Divisor Function

Olivier Bordellès
2, allée de la Combe
43000 Aiguilhe


Let $ q > 2$ be a prime number and define $ \lambda_q := \left(
\frac{\tau}{q} \right)$ where $ \tau(n)$ is the number of divisors of $ n$ and $ \left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When $ \tau(n)$ is a quadratic residue modulo $ q$, then the convolution $ \left( \lambda_q \star \mathbf{1} \right) (n)$ could be close to the number of divisors of $ n$. The aim of this work is to compare the mean value of the function $ \lambda_q \star \mathbf{1}$ to the well known average order of $ \tau$. A bound for short sums in the case $ q=5$ is also given, using profound results from the theory of integer points close to certain smooth curves.

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(Concerned with sequences A000005 A008836 A091337.)

Received January 14 2017; revised versions received June 1 2017; June 26 2017. Published in Journal of Integer Sequences, July 1 2017.

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