SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS
Forum of Mathematics, Sigma, Tome 4 (2016)
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Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$ . A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.
@article{10_1017_fms_2016_6,
author = {KAISA MATOM\"AKI and MAKSYM RADZIWI{\L}{\L} and TERENCE TAO},
title = {SIGN {PATTERNS} {OF} {THE} {LIOUVILLE} {AND} {M\"OBIUS} {FUNCTIONS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {4},
year = {2016},
doi = {10.1017/fms.2016.6},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.6/}
}
TY - JOUR AU - KAISA MATOMÄKI AU - MAKSYM RADZIWIŁŁ AU - TERENCE TAO TI - SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS JO - Forum of Mathematics, Sigma PY - 2016 VL - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.6/ DO - 10.1017/fms.2016.6 LA - en ID - 10_1017_fms_2016_6 ER -
KAISA MATOMÄKI; MAKSYM RADZIWIŁŁ; TERENCE TAO. SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.6
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