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On the prime density of Lucas sequences

Pieter Moree — 1996

Journal de théorie des nombres de Bordeaux

The density of primes dividing at least one term of the Lucas sequence L n ( P ) n = 0 , defined by L 0 ( P ) = 2 , L 1 ( P ) = P and L n ( P ) = P L n - 1 ( P ) + L n - 2 ( P ) for n 2 , with P an arbitrary integer, is determined.

Prime divisors of the Lagarias sequence

Pieter MoreePeter Stevenhagen — 2001

Journal de théorie des nombres de Bordeaux

We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence x n n = 1 defined by the linear recurrence x n + 1 = x n + x n - 1 and the initial values x 0 = 3 and x 1 = 1 . This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.

On Robin’s criterion for the Riemann hypothesis

YoungJu ChoieNicolas LichiardopolPieter MoreePatrick Solé — 2007

Journal de Théorie des Nombres de Bordeaux

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 fifth power...

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