Given a sequence 𝒜=false{a00,
then there must exist 𝒜⊆double-struckN with false|𝒜∩false[0,xfalse]false|=normalΘfalse(ffalse(xfalse)false) for which r𝒜,h + ℓ(n) = Θ(f(n)h + ℓ/n) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to x1false/2normallogfalse(xfalse)1false/2≪ffalse(xfalse)≪x. The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.
Let q ≥ 2 be an integer, χ (mod q) a primitive Dirichlet character, and f : Z ≥2 → R a function satisfying 2 ≤ f (q) ≪ log(q). We show that, if L(s, χ) has no zeros in the regionuniformly for primitive χ (mod q). As an example of an application, we show that the uniform abc-conjecture implies a strong version of "no Siegel zeros" for odd real characters of q o(1) -smooth moduli, by using our result in T. [9] together with a theorem of Chang [1] on zero-free regions.
The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy & Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as "Cramér's model"-like and density-type arguments, both of which our heuristic draws from.Apart from the prime number theorem, our argument is entirely elementary. The idea is to model sums of two primes by a hypergeometric probability distribution and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.2010 Mathematics Subject Classification. 11P32.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.