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**1 - 2**of**2**### On Montgomery’s 13’th problem

, 2008

"... 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 improves on a bound of Erdős-Renyi by a factor of the order √ l ..."

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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 improves on a bound of Erdős-Renyi by a factor of the order √ log n. 1

### On some power sum problems of Montgomery and Turán

, 2008

"... 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 |zk|=1 maxν=1,...,nh | ∑n k=1 zν k | ≤ (h − 1 + o(1))√n. This gives the right order of magnitude for the quantity and improves on a bound of Er ..."

Abstract
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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 |zk|=1 maxν=1,...,nh | ∑n k=1 zν k | ≤ (h − 1 + o(1))√n. 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