Geodesic
Long and short sums of a twisted divisor function
Journal of integer sequences, Tome 20 (2017) no. 7
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.
Classification : 11N37, 11A25, 11M41
Keywords: number of divisors, Legendre symbol, mean value, Riemann hypothesis 
@article{JIS_2017__20_7_a4,
     author = {Bordell\`es,  Olivier},
     title = {Long and short sums of a twisted divisor function},
     journal = {Journal of integer sequences},
     year = {2017},
     volume = {20},
     number = {7},
     zbl = {1422.11196},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_7_a4/}
}
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Bordellès,  Olivier. Long and short sums of a twisted divisor function. Journal of integer sequences, Tome 20 (2017) no. 7. http://geodesic.mathdoc.fr/item/JIS_2017__20_7_a4/