On the Möbius function

Abstract

We investigate incomplete convolutions of the Möbius function of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript d vertical-bar n semicolon d less-than-or-equal-to z Endscripts mu left-parenthesis d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo movablelimits="false">∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>;</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum \nolimits _{d|n;d \leq z} {\mu (d)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that for almost all integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> one can find <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z"> <mml:semantics> <mml:mi>z</mml:mi> <mml:annotation encoding="application/x-tex">z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which this sum is large.

Locations

• Transactions of the American Mathematical Society - View - PDF