×

Congruences and exponential sums with the sum of aliquot divisors function. (English) Zbl 1231.11094

Summary: We give bounds on the number of integers \(1\leq n\leq N\) such that \(p\mid s(n)\), where \(p\) is a prime and \(s(n)\) is the sum of aliquot divisors function given by \(s(n) = \sigma(n) - n\), where \(\sigma(n)\) is the sum of divisors function. Using this result, we obtain nontrivial bounds in certain ranges for rational exponential sums of the form \[ S_p(a,N)=\sum_{n\leq N} \exp(2\pi ias(n)/p),\quad \gcd(a,p)=1. \]

MSC:

11L07 Estimates on exponential sums
11A07 Congruences; primitive roots; residue systems
11N60 Distribution functions associated with additive and positive multiplicative functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] W. Banks and I. E. Shparlinski, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications 41, eds. A. van der Poorten and A. Stein (Amer. Math. Soc., 2004) pp. 49–60. · doi:10.1090/fic/041/04
[2] DOI: 10.1216/rmjm/1181069373 · Zbl 1193.11095 · doi:10.1216/rmjm/1181069373
[3] DOI: 10.1016/0022-314X(83)90002-1 · Zbl 0513.10043 · doi:10.1016/0022-314X(83)90002-1
[4] DOI: 10.1007/978-3-662-04658-6 · doi:10.1007/978-3-662-04658-6
[5] Halberstam H., Sieve Methods (1974)
[6] DOI: 10.5802/jtnb.101 · Zbl 0797.11070 · doi:10.5802/jtnb.101
[7] Vaughan R. C., J. London Math. Soc. 10 pp 153–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.