The index of a Farey sequence

Abstract

We define an integer valued function of the Farey fractions which we call the index, and we prove two exact formulae involving this. We give asymptotic formulae for the second moment of the index and for the value distribution. Zagier communicated to us a remarkable formula relating the index to Dedekind sums and this yields further asymptotic formulae. 2000 Mathematics Subject Classification: 11B57, 11F20, 11L07 R. R. Hall Department of Mathematics York University Heslington York YO10 5DD United Kingdom Email: [email protected] P. Shiu Department of Mathematical Sciences Loughborough University Loughborough Leicestershire LE11 3TU United Kingdom Email: [email protected] The index of a Farey sequence R. R. Hall and P. Shiu Abstract We define an integer valued function of the Farey fractions which we call the index, and we prove two exact formulae involving this. We give asymptotic formulae for the second moment of the index and for the value distribution. Zagier communicated to us a remarkable formula relating the index to Dedekind sums and this yields further asymptotic formulae.

Locations

• The Michigan Mathematical Journal - View - PDF