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An application of Ramanujan sums to equirepartition modulo an odd integer

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Author(s): Éric Balandraud

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 2;
Issue: 2;
Start page: 1;
Date: 2007;
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Keywords: Ramanujan sum | Euler totient | Jacobi symbol | primitive root of unity

ABSTRACT
Following a result of Myerson on finite fields,we prove that for an odd number $n\geqslant 3$,the $2^{ \phi(n)/2}$ sums:$$ \pm 1 \pm 2 \dots \pm i \dots \pm (n-1)/2,$$where the terms $i$ are all the invertiblesmodulo $n$ from $1$ to $(n-1)/2,$ are distributedamong the classes modulo $n$ as uniformly as possible. \parWe also determine and prove that the distribution of the$2^{(n-1)/2}$ sums:$$ \pm 1 \pm 2 \dots \pm (n-1)/2,$$where all the elements from $1$ to $(n-1)/2$ are represented once,is also asymptotically equidistributed.
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