Restriction theory of the Selberg sieve, with applications

Ben Green; Terence Tao

Displaying similar documents to “Restriction theory of the Selberg sieve, with applications”

On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

Maurizio Laporta (2012)

Journal de Théorie des Nombres de Bordeaux

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We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N = p_{1}^{g} + p_{2}^{g} + ... + p_{s}^{g}$ with $p_{1}, p_{2}, ..., p_{s}$ prime numbers such that $p_{1} \equiv l (\mod k)$ , under suitable hypothesis on $s = s (g)$ for every integer $g \geq 2$ .

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.

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

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Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2 (mod p)$ is odd; it is $7 / 24$ . Here we mimic successfully Hasse’s method to compute the density $δ_{q}$ of monic irreducibles $P$ in $𝔽_{q} [X]$ for which the order of $X (mod P)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the $δ_{p}$ ’s as $p$ varies through all rational primes.

On the Distribution of M-tuples of B-numbers

Werner Georg Nowak (2005)

Publications de l'Institut Mathématique

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Landau’s function for one million billions

Marc Deléglise, Jean-Louis Nicolas, Paul Zimmermann (2008)

Journal de Théorie des Nombres de Bordeaux

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Let $𝔖_{n}$ denote the symmetric group with $n$ letters, and $g (n)$ the maximal order of an element of $𝔖_{n}$ . If the standard factorization of $M$ into primes is $M = q_{1}^{α_{1}} q_{2}^{α_{2}} ... q_{k}^{α_{k}}$ , we define $ℓ (M)$ to be $q_{1}^{α_{1}} + q_{2}^{α_{2}} + ... + q_{k}^{α_{k}}$ ; one century ago, E. Landau proved that $g (n) = {max}_{ℓ (M) \leq n} M$ and that, when $n$ goes to infinity, $log g (n) \sim \sqrt{n log (n)}$ . There exists a basic algorithm to compute $g (n)$ for $1 \leq n \leq N$ ; its running time is $𝒪 (N^{3 / 2} / \sqrt{log N})$ and the needed memory is $𝒪 (N)$ ; it allows computing $g (n)$ up to, say, one million. We describe an algorithm to calculate $g (n)$ for $n$ up to $10^{15}$ . The main idea is to use the...

Displaying similar documents to “Restriction theory of the Selberg sieve, with applications”

On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

On rough and smooth neighbors.

Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

On the Distribution of M-tuples of B-numbers

Landau’s function for one million billions

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd