On Montgomery's 13th problem
Johan Andersson · arXiv (Cornell University) · 2007
We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turán. Let h ≥ 2 be an integer. We prove that inf max n∑ z ν k ∣ ≤ (h − 1)√n + O ( n 0.2625+ǫ). (ǫ> 0) |zk|≥1 ν=1,...,nh k=1 This gives the right order of magnitude for the quantity and improves on a bound of Erdős-Renyi by a factor of the order √ log n. 1