Correlations of multiplicative functions and applications

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Date(s) - 06/12/2016
14 h 00 min - 15 h 00 min

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We develop the asymptotic formulas for correlations
Σ{n}≤{x}{f}1({P}1({n})) {f}2({P}2({n})) … {f}{m}({P}{m}({n}))
where {f}1, …, {f}{m} are bounded « pretentious » multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences: first, we characterize all multiplicative functions {f}: {{N}} → {-1,+1} with bounded partial sums. This answers a question of Erdős from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either {f}({n})={n}{s} for Re({s})<1 or |{f}({n})| is small on average. This settles an old conjecture of Kátai. Third, we discuss some recent applications to the study of sign patterns of ({f}({n}),{f}({n}+1),{f}({n}+2)) and ({f}({n}),{f}({n}+1),{f}({n}+2),{f}({n}+3))} where {f}: {{N}} → {-1,+1} is a given multiplicative function. If time permits, we discuss multidimensional version of some of the results mentioned above.


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