Geodesic
Extremal Values of
Canadian mathematical bulletin, Tome 41 (1998) no. 3, pp. 335-347

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DOI

The function $\Delta (x,N)$ as defined in the title is closely associated via $\Delta \,(N)\,=\,{{\sup }_{x}}\,|\,\Delta (x,N)|$ to several problems in the upper bound sieve. It is also known via a classical theorem of Franel that certain conjectured bounds involving averages of $\Delta (x,N)$ are equivalent to the Riemann Hypothesis. We improve the unconditional bounds which have been hitherto obtained for $\Delta (N)$ and show that these are close to being optimal. Several auxiliary results relating $\Delta (Np)$ to $\Delta (N)$ , where $p$ is a prime with $p\nmid N$ , are also obtained and two new conjectures stated.
DOI : 10.4153/CMB-1998-046-3
Mots-clés : 11N25 
Codecà, P.; Nair, M. Extremal Values of. Canadian mathematical bulletin, Tome 41 (1998) no. 3, pp. 335-347. doi: 10.4153/CMB-1998-046-3
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     title = {Extremal {Values} of},
     journal = {Canadian mathematical bulletin},
     pages = {335--347},
     year = {1998},
     volume = {41},
     number = {3},
     doi = {10.4153/CMB-1998-046-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-046-3/}
}
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