Mean value of Piltz' function over integers free of large prime factors.
Nyandwi, Servat (2003)
Publications de l'Institut Mathématique. Nouvelle Série
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Nyandwi, Servat (2003)
Publications de l'Institut Mathématique. Nouvelle Série
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Harold G. Diamond, Janos Pintz (2009)
Journal de Théorie des Nombres de Bordeaux
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Mertens’ product formula asserts that as . Calculation shows that the right side of the formula exceeds the left side for . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on , this and a complementary inequality might change their sense for sufficiently large values of . We show this to be the case.
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé (2007)
Journal de Théorie des Nombres de Bordeaux
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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality is satisfied for , where denotes the Euler(-Mascheroni) constant. We show by elementary methods that if does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that must be divisible by a fifth power . As consequence we obtain that RH holds true iff every natural number divisible by...
Florian Luca, Paul Pollack (2011)
Journal de Théorie des Nombres de Bordeaux
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Let denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every , the equation has a solution . This suggests defining as the number of solutions to the equation . (So Carmichael’s conjecture asserts that always.) Results on are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of contains every natural number . Also, the maximal order of has been investigated by Erdős and Pomerance....
Luis J. Boya (2008)
RACSAM
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Robert Černý, Silvie Mašková (2010)
Czechoslovak Mathematical Journal
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Let be a bounded open set in , . In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of such that We extend this result to the situation in which the underlying space is replaced by the generalized Zygmund space , the corresponding space of exponential growth then being given by a Young function which behaves like for large . We also discuss the case of an embedding into triple and other multiple exponential cases.