Group items tagged

Gergely Harcos The precise bound was first proved by Wigert in 1906 using the prime number theorem, while Ramanujan in 1914 observed its elementary character. In fact we can prove the inequality even without knowing unique factorization! All we need to know is that and imply . This property implies as one can inject the set of divisors of into the set of pairs formed of a divisor of and a divisor of : to assign the pair . Once we know we can see for any positive integer that . It follows that , whence also . Now the second exponent changes by a factor less than 2 whenever is increased by 1, so we can certainly find a with . This choice furnishes Wigert’s estimate upon observing that .

We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.