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Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux
"... Abstract. 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 ktuples. Let a1,..., ak and b1,. ..."
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Abstract. 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 ktuples. Let a1,..., ak and b1,...,bk be positive integers. Write h(θ): = ∑ n∈X e(nθ), where X is the set of all n � N such that the numbers a1n + b1,..., akn + bk are all prime. We obtain upper bounds for ‖h ‖ L p (T), p> 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
Journal de Théorie des Nombres
, 2005
"... Restriction theory of the Selberg sieve, with applications ..."
Series involving Arithmetic Functions
, 2007
"... We intend here to collect infinite series, each involving unusual combinations or variations of wellknown arithmetic functions. For simplicity’s sake, results are often quoted not with full generality but only to illustrate a special case. Let σ(n) denote the sum of all distinct divisors of n, κ(n) ..."
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We intend here to collect infinite series, each involving unusual combinations or variations of wellknown arithmetic functions. For simplicity’s sake, results are often quoted not with full generality but only to illustrate a special case. Let σ(n) denote the sum of all distinct divisors of n, κ(n) denote the quotient of n with its greatest square divisor, and ϕ(n) denote the number of positive integers k ≤ n satisfying gcd(k, n) = 1. These multiplicative functions are called sumofdivisors, squarefree part, and Euler totient, respectively. It can be shown that the following series are convergent: ∞X n=1