A Variant of Möbius Inversion

ABSTRACT

We develop a new approach to Möbius inversion by considering the spectral problem for certain symmetric, positive-definite matrices closely connected to the divisibility relation. This spectral problem appears to have significance for analytic number theory. We obtain a novel version of the second Möbius inversion formula, and in particular a new formula for the Mertens function. Computational evidence regarding the latter formula may throw some light on the Riemann Hypothesis.

KEYWORDS:

• Mertens function
• Riemann hypothesis
• Möbius inversion
• divisor matrix

2010 AMS SUBJECT CLASSIFICATION:

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