Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

2014/12/01 by Roger C. Baker, Liangyi Zhao, Baker, Roger C. +1
Mathematics · #11N13 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1412.0574

openalex publication_date 2014/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let t ∈ ℕ, η>0. Suppose that x is a sufficiently large real number and q is a natural number with q ≤ x^5/12-η, q not a multiple of the conductor of the exceptional character χ^* (if it exists). Suppose further that, max \p : p | q \ lt; exp ((log x)/(C log log x)) and ∏_{p | q} p lt; x^δ, where C and δ are suitable positive constants depending on t and η. Let a ∈ ℤ, (a,q)=1 and A = \n ∈ (x/2, x]: n ≡ a \pmodq \ . We prove that there are primes p₁ < p₂ < ... < p_t in A with p_t - p₁ ≪ qt exp ((40 t)/(9-20 θ)) . Here θ= (log q) / log x.

Citations

• Prime Numbers in Short Intervals and a Generalized Vaughan Identity

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