On Bombieri's asymptotic sieve

On Bombieri’s Asymptotic Sieve

If a sequence (an) of non-negative real numbers has “best possible” distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum ∑ n≤x anΛk(n) for k ≥ 2. By constructing appropriate sequences, we show that any weakening of the welldistribution property is not sufficient to deduce the same conclusion.

Asymptotic sieve for primes

For a long time it was believed that sieve methods might be incapable of achieving the goal for which they had been created, the detection of prime numbers. Indeed, it was shown [S], [B] to be inevitable that, in the sieve’s original framework, no such result was possible although one could come tantalizingly close. This general limitation is recognized as the “parity problem” of sieve theory. ...

Notes on the large sieve

On the Square-free Sieve

where the main term will depend on the application; in general, the error term will depend only on the polynomial P in question, not on the particular quantity being estimated. We can split the error term further into one term that can be bounded for every given P , and a second term, say, δ(P ), which may be rather hard to estimate, and which is unknown for polynomials P of high enough degree....

Asymptotic analysis of the sieve estimator for a class of parabolic SPDEs

In this paper we consider the problem of estimating a coefficient of a strongly elliptic partial differential operator in stochastic parabolic equations. The coefficient is a bounded function of time. We compute the maximum likelihood estimate of the function on an approximating space (sieve) using a finite number of the spatial Fourier coefficients of the solution and establish conditions that...