Journal Article

Arithmetic Correlations Over Large Finite Fields

Jonathan P. Keating and Edva Roditty-Gershon

in International Mathematics Research Notices

Volume 2016, issue 3, pages 860-874
Published in print January 2016 | ISSN: 1073-7928
Published online June 2015 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1093/imrn/rnv157
Arithmetic Correlations Over Large Finite Fields

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The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M[math] bius function, and the divisor function, are the subject of classical problems in analytic number theory. The function field analogs of these problems have recently been resolved in the limit of large finite field size [math]. However, in this limit, the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in [math] which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when [math]; in particular, one cannot expect remainder terms that are of the order of the square-root of the main term in this context.

Journal Article.  3174 words. 

Subjects: Mathematics ; Pure Mathematics

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