Abstract
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors’ new proof of Halász’s theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.
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References
Elliott, PDTA: Extrapolating the mean-values of multiplicative functions. Nederl. Akad. Wetensch. Indag. Math. 51(4), 409–420 (1989).
Granville, A, Harper, AJ, Soundararajan, K: A new proof of Halász’s theorem, and its consequences. In preparation.
Granville, A, Soundararajan, K: The spectrum of multiplicative functions. Ann. Math. (2). 153(2), 407–470 (2001).
Granville, A, Soundararajan, K: Decay of mean values of multiplicative functions. Canad. J. Math. 55(6), 1191–1230 (2003).
Koukoulopoulos, D: On multiplicative functions which are small on average. Geom. Funct. Anal. 23(5), 1569–1630 (2013).
Rosen, M: Number theory in function fields. Graduate Texts in Mathematics. New York. 210 (2002).
Acknowledgement
Andrew Granville’s research is partly supported by NSERC, Canada, and a Canadian Research Chair. Adam Harper is supported by a research fellowship at Jesus College, Cambridge. Kannan Soundararajan was partially supported by NSF grant DMS 1001068, and a Simons Investigator grant from the Simons Foundation.
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Granville, A., Harper, A.J. & Soundararajan, K. Mean values of multiplicative functions over function fields. Res. number theory 1, 25 (2015). https://doi.org/10.1007/s40993-015-0023-5
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DOI: https://doi.org/10.1007/s40993-015-0023-5