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Search: MSC category 11N60 ( Distribution functions associated with additive and positive multiplicative functions )

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1. CJM 2009 (vol 61 pp. 481)

Banks, William D.; Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E.
Uniform Distribution of Fractional Parts Related to Pseudoprimes
We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and $P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$ and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base $g$, and if $n$ is a Carmichael number, this is true for all $g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We also obtain similar results with the functions ${\widetilde f}_g(n) = gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.

Categories:11L07, 11N37, 11N60

2. CJM 2003 (vol 55 pp. 1191)

Granville, Andrew; Soundararajan, K.
Decay of Mean Values of Multiplicative Functions
For given multiplicative function $f$, with $|f(n)| \leq 1$ for all $n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq x} f(n)$ converges. Hal\'asz showed that this depends on the minimum $M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1 - \Re (f(p) p^{-iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll (1+M) e^{-M}$. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Hal\'asz-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of $y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the $k$-th powers mod~$p$.

Categories:11N60, 11N56, 10K20, 11N37

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