Journal of the Australian Mathematical Society

Research Article

BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

MICHAEL J. MOSSINGHOFFa1 c1 and TIMOTHY S. TRUDGIANa2

a1 Department of Mathematics, Davidson College, Davidson, NC, 28035-6996, USA (email: mimossinghoff@davidson.edu)

a2 Department of Mathematics and Computer Science, University of Lethbridge, Alberta, Canada T1K 3M4 (email: tim.trudgian@uleth.ca)

Abstract

We investigate the behaviour of the function , where is the Liouville function and is a real parameter. The case where was investigated by Pólya; the case , by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family , where , and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where is of particular interest.

(Received May 06 2011)

(Accepted February 01 2012)

(Online publication September 27 2012)

2010 Mathematics subject classification

  • primary 11M26; secondary 11M45;
  • 11N64;
  • 11Y35

Keywords and phrases

  • Liouville function;
  • Riemann hypothesis;
  • Pólya’s problem;
  • Turán’s problem

Correspondence

c1 For correspondence; e-mail: mimossinghoff@davidson.edu

Footnotes

  Dedicated to the memory of Alf van der Poorten

  Communicated by I. E. Shparlinski

  This work was partially supported by a grant from the Simons Foundation (#210069 to M. Mossinghoff).