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We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near σ = 1 " for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 is an integer, A ⊆ { 1 , ... , N } , and for all primes p no two elements in A differ by p - 1 , then | A | ≪ N 1 - 10 - 18 . This strengthens a theorem of Sárközy. [ABSTRACT FROM AUTHOR]

Let k be a number field and G be a finite group. Let F k G (Q) be the family of number fields K with absolute discriminant D K at most Q such that K / k is normal with Galois group isomorphic to G . If G is the symmetric group S n or any transitive group of prime degree, then we unconditionally prove that for all K ∈ F k G (Q) with at most O ε (Q ε) exceptions, the L -functions associated to the faithful Artin representations of Gal (K / k) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree n S n -fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; for a prime p , the periodic torus orbits attached to the ideal classes of almost all totally real degree p fields F over ℚ equidistribute on PGL p (Z) ∖ PGL p (R) with respect to Haar measure; for each ℓ ≥ 2 , the ℓ -torsion subgroups of the ideal class groups of almost all degree p fields over k (resp. almost all degree n S n -fields over k ) are as small as GRH implies; and an effective variant of the Chebotarev density theorem holds for almost all fields in such families. [ABSTRACT FROM AUTHOR]

Let K be a number field and, for an integral ideal q of K , let χ be a character of the narrow ray class group modulo q . We establish various new and improved explicit results, with effective dependence on K , q and χ , regarding the zeros of the Hecke L -function L ( s , χ ) , such as zero-free regions, Deuring–Heilbronn phenomenon, and zero density estimates. [ABSTRACT FROM AUTHOR]

Let F n be the set of cuspidal automorphic representations π of GL n over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of π ∈ F n , one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of L -functions L (s , π) associated to π ∈ F n , which we make log-free. As an application of the zero density estimate, we prove a hybrid subconvexity bound for L (1 2 , π) for a density one subset of π ∈ F n. [ABSTRACT FROM AUTHOR]