A note on the Selberg Sieve and the large sieve

Notes on the large sieve

Note on the Consistency of Sieve Estimators

In this note the consistency of sieve estimators is derived without requiring compactness of the entire parameter space, and allowing the expected objective function to take infinite values.

On a Variant of the Large Sieve

We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N ] = {1, . . . , N} and, for each odd prime p 6 √ N , remove or “sieve out” by all n whose reduction n(mod p) lies in some interval Ip ⊆ Z/pZ of length (p−1)/2. Let A be the set that remains: then |A| ≪ N log N , a bound which improves slightly on the bound of |A| ≪ N logN which re...

Restriction Theory of the Selberg Sieve, with Applications

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L–L restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a1, . . . , ak and b1, . . . , bk be positive integers. Write h(θ) := ∑ n∈X e(nθ), where X is the set of all n 6 N such that the nu...

On the large sieve with square moduli

We prove an estimate for the large sieve with square moduli which improves a recent result of L. Zhao. Our method uses an idea of D. Wolke and some results from Fourier analysis. Mathematics Subject Classification (2000): 11N35, 11L07, 11B57