Duke Mathematical Journal
- Duke Math. J.
- Volume 149, Number 1 (2009), 1-36.
The hyperbolic lattice point count in infinite volume with applications to sieves
We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms that are uniform as the lattice moves through “congruence” subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, , but restrict to the orbit , where is an infinite-index, nonelementary, finitely generated subgroup of . Assume that the Reimann surface has a cusp at infinity. We show that the set of values contains infinitely many integers having at most prime factors for any , where is the spectral gap and is the Hausdorff dimension of the limit set of . If , then we can take , giving . The limit of this method is for . This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture
Duke Math. J., Volume 149, Number 1 (2009), 1-36.
First available in Project Euclid: 1 July 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]
Secondary: 11F72: Spectral theory; Selberg trace formula 11N36: Applications of sieve methods
Kontorovich, Alex V. The hyperbolic lattice point count in infinite volume with applications to sieves. Duke Math. J. 149 (2009), no. 1, 1--36. doi:10.1215/00127094-2009-035. https://projecteuclid.org/euclid.dmj/1246453788