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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.doi:10.24033/bsmf.2609 fatcat:vr2moqef6fdj3hmermys6qwagu