Geodesic
Links between $\Delta(x,N) = {\displaystyle \sum_{{n \leq xN, \,\, (n,N)=1}}} 1-x\phi(N)$ and character sums
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 509-516

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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 $s$. 
@article{BUMI_2003_8_6B_2_a14,
     author = {Codec\'a, P. and Nair, M.},
     title = {Links between $\Delta(x,N) = {\displaystyle \sum_{{n \leq xN, \,\, (n,N)=1}}} 1-x\phi(N)$ and character sums},
     journal = {Bollettino della Unione matematica italiana},
     pages = {509--516},
     publisher = {mathdoc},
     volume = {Ser. 8, 6B},
     number = {2},
     year = {2003},
     zbl = {1177.11079},
     mrnumber = {MR1988219},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_2_a14/}
}
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Codecá, P.; Nair, M. Links between $\Delta(x,N) = {\displaystyle \sum_{{n \leq xN, \,\, (n,N)=1}}} 1-x\phi(N)$ and character sums. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 509-516. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_2_a14/