We prove that the sign of Kloosterman sums changes infinitely often as runs through the square-free numbers with at most 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.
Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by . The system is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to . This paper is a remark to Rutkowski’s paper. We define another system in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...
A strongly pseudoconvex function is generalized to non-smooth settings. A complete characterization of the strongly pseudoconvex radially lower semicontinuous functions is obtained.
We improve the constants in the Men’shov-Rademacher inequality by showing that for n ≥ 64, for all orthogonal random variables X₁,..., Xₙ such that .
The Poincaré inequality is extended to uniformly doubling metric-measure spaces which satisfy a version of the triangle comparison property. The proof is based on a generalization of the change of variables formula.
In this note a uniform transparent presentation of the scalar Haffian will be given. Some well-known results will be generalized. A link will be established between the scalar Haffian and the derivative matrix as developed by Magnus and Neudecker.
We improve a theorem of C. L. Belna (1972) which concerns boundary behaviour of complex-valued functions in the open upper half-plane and gives a partial answer to the (still open) three-segment problem.