# 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

Bollettino dell'Unione Matematica Italiana (2003)

- Volume: 6-B, Issue: 2, page 509-516
- ISSN: 0392-4041

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topCodecá, 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

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

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