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Define to be the complexity of , the smallest number of 1s needed to write using an arbitrary combination of addition and multiplication. John Selfridge showed that for all , leading this author and Zelinsky to define the defect of , , to be the difference . Meanwhile, in the study of addition chains, it is common to consider , the number of small steps of , defined as , an integer quantity, where is the length of the shortest addition chain for . So here we analogously define , the integer defect of , an integer version of analogous to . Note that is not the same as .
We show that has additional meaning in terms of the defect well-ordering we considered in 2015, in that indicates which powers of the quantity lies between when one restricts to with lying in a specified congruence class modulo . We also determine all numbers with , and use this to generalize a result of Rawsthorne (1989).
In order to analyse the simultaneous approximation properties of reals, the parametric geometry of numbers studies the joint behaviour of the successive minima functions with respect to a one-parameter family of convex bodies and a lattice defined in terms of the given reals. For simultaneous approximation in the sense of Dirichlet, the linear independence over of these reals together with 1 is equivalent to a certain nice intersection property that any two consecutive minima functions enjoy. This paper focusses on a slightly generalized version of simultaneous approximation where this equivalence is no longer in place and investigates conditions for that intersection property in the case of linearly dependent irrationals.
We continue our investigations regarding the distribution of positive and negative values of Hardy’s -functions in the interval when the conductor and both tend to infinity. We show that for , , with , satisfying , the Lebesgue measure of the set of values of for which is as , where denotes the number of distinct prime factors of the conductor of the character , and is the usual Euler totient. This improves upon our earlier result. We also include a corrigendum for the first part of this article.
We study the mean square of the error term in the Gauss circle problem. A heuristic argument based on the consideration of off-diagonal terms in the mean square of the relevant Voronoi-type summation formula leads to a precise conjecture for the mean square of this discrepancy.
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