Displaying similar documents to “Oscillation of Mertens’ product formula”

Perfect powers in the summatory function of the power tower

Florian Luca, Diego Marques (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let ( a n ) n 1 be the sequence given by a 1 = 1 and a n = n a n - 1 for n 2 . In this paper, we show that the only solution of the equation a 1 + + a n = m l is in positive integers l > 1 , m and n is m = n = 1 .

An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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....

On Robin’s criterion for the Riemann hypothesis

YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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...

Explicit lower bounds for linear forms in two logarithms

Nicolas Gouillon (2006)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in []. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around 5 . 10 4 instead of 10 8 .