An optimization problem concerning multiplicative functions

Article English OPEN
Hilberdink, Titus (2015)

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0<f(p)<1, the solution is at a multiplicative point for all q≥1.
  • References (8)

    [1] C. Aistleitner, I. Berkes, and K. Seip, GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc. (to appear). (See arXiv:1210.0741.)

    [2] A. Bondarenko and K. Seip, GCD sums and complete sets of square-free numbers, Bull. London Math. Soc. 47 (2015) 29-41.

    [3] P. Codeca and M. Nair, Calculating a Determinant associated with Multiplicative Functions, Bollettino Unione Math. Ital. (8) 5 (2002) 545-555.

    [4] T. Dyer and G. Harman, Sums involving common divisors, J. London Math. Soc. 34 (1986) 1-11.

    [5] I. S. Gal, A theorem concerning Diophantine approximations, Nieuw Arch. Wiskunde 23 (1949) 13-38.

    [6] T. W. Hilberdink, The group of squarefree integers, Linear Algebra and its Applications 457 (2014) 383-399.

    [7] A. Perelli and U. Zannier, An extremal property of the Mo&#x7f;bius function, Arch. Math. 53 (1989) 20-29.

    [8] A. E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, 1958.

  • Metrics
    0
    views in OpenAIRE
    0
    views in local repository
    14
    downloads in local repository

    The information is available from the following content providers:

    From Number Of Views Number Of Downloads
    Central Archive at the University of Reading - IRUS-UK 0 14
Share - Bookmark