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A note on signs of Kloosterman sums

Kaisa Matomäki — 2011

Bulletin de la Société Mathématique de France

We prove that the sign of Kloosterman sums Kl ( 1 , 1 ; n ) changes infinitely often as n runs through the square-free numbers with at most 15 prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

Sums of positive density subsets of the primes

Kaisa Matomäki — 2013

Acta Arithmetica

We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least ( 1 - o ( 1 ) ) α / ( e γ l o g l o g ( 1 / β ) ) , which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of * m using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any A , B * m of densities α and β, the...

Real zeros of holomorphic Hecke cusp forms and sieving short intervals

Kaisa Matomäki — 2016

Journal of the European Mathematical Society

We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.

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