# Links between $\mathrm{\Delta }\left(x,N\right)=\sum _{n\le xN,\left(n,N\right)=1}1-x\varphi \left(N\right)$ and character sums

• Volume: 6-B, Issue: 2, page 509-516
• ISSN: 0392-4033

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## Abstract

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We express $\mathrm{\Delta }\left(x,N\right)$, as defined in the title, for $x=\frac{a}{q}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\mathrm{\Delta }\left(N\right)={sup}_{x\in \mathbb{R}}\left|\mathrm{\Delta }\left(x,N\right)\right|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.

## How to cite

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Codecá, P., and Nair, M.. "Links between $\Delta(x,N) = {\displaystyle \sum_{{n \leq xN, \,\, (n,N)=1}}} 1-x\phi(N)$ and character sums." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 509-516. <http://eudml.org/doc/195953>.

@article{Codecá2003,
abstract = {We express $\Delta(x, N)$, as defined in the title, for $x=\frac\{a\}\{q\}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\Delta(N)= \sup_\{x\in \mathbb\{R\}\} |\Delta (x,N)|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.},
author = {Codecá, P., Nair, M.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {509-516},
publisher = {Unione Matematica Italiana},
title = {Links between $\Delta(x,N) = \{\displaystyle \sum_\{\{n \leq xN, \,\, (n,N)=1\}\}\} 1-x\phi(N)$ and character sums},
url = {http://eudml.org/doc/195953},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Codecá, P.
AU - Nair, M.
TI - Links between $\Delta(x,N) = {\displaystyle \sum_{{n \leq xN, \,\, (n,N)=1}}} 1-x\phi(N)$ and character sums
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 509
EP - 516
AB - We express $\Delta(x, N)$, as defined in the title, for $x=\frac{a}{q}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\Delta(N)= \sup_{x\in \mathbb{R}} |\Delta (x,N)|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.
LA - eng
UR - http://eudml.org/doc/195953
ER -

## References

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1. CODECÀ, P.- NAIR, M., Extremal Values of $\mathrm{\Delta }\left(x,N\right)=\sum _{\begin{array}{c}n\le xN\\ \left(n,N\right)=1\end{array}}1-x\varphi \left(N\right)$. Canad. Math. Bull. Vol., 41 (3), (1998), 335-347. Zbl0920.11066MR1637673
2. INGHAM, A. E., The Distribution of Prime Numbers, Hafner Publishing Company, New York1971. Zbl0006.39701MR184920JFM58.0193.02
3. WASHINGTON, L. C., Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, Springer-Verlag, New York1982. Zbl0484.12001MR718674

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