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Displaying similar documents to “Landau’s problems on primes”

Restriction theory of the Selberg sieve, with applications

Ben Green, Terence Tao (2006)

Journal de Théorie des Nombres de Bordeaux

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The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 – L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k -tuples. Let a 1 , ⋯ , a k and b 1 , ⋯ , b k be positive integers. Write h ( θ ) : = ∑ n ∈ X e ( n θ ) , where X is the set of all n ≤ N such that the numbers a 1 n + b 1 , ⋯ , a k n + b k are all prime. We obtain upper bounds for ∥ h ∥ L p ( 𝕋 ) , p > 2 , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct...

On rough and smooth neighbors.

William D. Banks, Florian Luca, Igor E. Shparlinski (2007)

Revista Matemática Complutense

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We study the behavior of the arithmetic functions defined by F(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1) where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.

Infinite families of noncototients

A. Flammenkamp, F. Luca (2000)

Colloquium Mathematicae

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For any positive integer n let ϕ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ( 2 m k ) m ≥ 1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

.121221222... is not quadratic.

Florian Luca (2005)

Revista Matemática Complutense

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In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑ (a / b), where a ∈ Z and 1 ≤ |a| ≤ K for all n ≥ 0, is neither rational nor quadratic.