Perfect powers in the summatory function of the power tower

Florian Luca; Diego Marques

Displaying similar documents to “Perfect powers in the summatory function of the power tower”

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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Oscillation of Mertens’ product formula

Harold G. Diamond, Janos Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

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Mertens’ product formula asserts that $\prod_{p \leq x} (1 - \frac{1}{p}) log x \to e^{- γ}$ as $x \to \infty$ . Calculation shows that the right side of the formula exceeds the left side for $2 \leq x \leq 10^{8}$ . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $π (x) - li x$ , this and a complementary inequality might change their sense for sufficiently large values of $x$ . We show this to be the case.

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 $σ (n) : = \sum_{d | n} d < e^{γ} n log log n$ is satisfied for $n \geq 5041$ , where $γ$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n \geq 37$ 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...

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 $m \neq n$ . This suggests defining $F (n)$ as the number of solutions $m$ to the equation $φ (n) = φ (m)$ . (So Carmichael’s conjecture asserts that $F (n) \geq 2$ 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 $k \geq 2$ . Also, the maximal order of $F$ has been investigated by Erdős and Pomerance....

Another relation between π, e, γ and ζ(n).

Luis J. Boya (2008)

RACSAM

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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 $ℝ^{n}$ , $n \geq 2$ . In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that $sup \{\int_{Ω} exp ({(\frac{|f (x)|}{K})}^{n / (n - 1)}) : f \in W_{0}^{1, n} (Ω), {∥ \nabla f ∥}_{L^{n}} \leq 1\} < \infty .$ 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.