Displaying similar documents to “On Robin’s criterion for the Riemann hypothesis”

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....

Palindromic powers.

Hernández, Santos Hernández, Luca, Florian (2006)

Revista Colombiana de Matemáticas

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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.