Displaying similar documents to “A variational method for univalent functions connected with antigraphy”

On Robin’s criterion for the Riemann hypothesis

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 σ ( n ) : = ∑ d | n d < e γ n log log n is satisfied for n ≥ 5041 , where γ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n ≥ 37 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 n must be divisible by a fifth power > 1 . As consequence we obtain that RH holds true iff every natural number divisible by...

A sharp form of an embedding into multiple exponential spaces

Robert Černý, Silvie Mašková (2010)

Czechoslovak Mathematical Journal

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Let Ω be a bounded open set in ℝ n , n ≥ 2 . In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of K such that sup ∫ Ω exp f ( x ) K n / ( n - 1 ) : f ∈ W 0 1 , n ( Ω ) , ∥ ∇ f ∥ L n ≤ 1 < ∞ . We extend this result to the situation in which the underlying space L n is replaced by the generalized Zygmund space L n log n - 1 L log α log L ( α < n - 1 ) , the corresponding space of exponential growth then being given by a Young function which behaves like exp ( exp ( t n / ( n - 1 - α ) ) ) for large t . We also discuss the case of an embedding into triple and other multiple exponential cases.

Palindromic powers.

Hernández, Santos Hernández, Luca, Florian (2006)

Revista Colombiana de Matemáticas

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An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

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Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n , the equation φ ( n ) = φ ( m ) has a solution m ≠ n . This suggests defining F ( n ) as the number of solutions m to the equation φ ( n ) = φ ( m ) . (So Carmichael’s conjecture asserts that F ( n ) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of F contains every natural number k ≥ 2 . Also, the maximal order of F has been investigated by Erdős and Pomerance....