Sums of singular series and primes in short intervals in algebraic number fields

The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero

“…In this paper we show that the answer to both of these questions is affirmative assuming a suitably strong version of the Hardy-Littlewood prime tuples conjecture [10]. We use a (very slightly strengthened) formulation from the recent paper [11]:…”

“…It is tempting to use Gallagher type estimates [6] on the mean value of singular series to control the left-hand side (which would morally obtain the sharper bound 𝑂(𝑒 −2𝜆 )), but unfortunately the best known bounds, see [11,Theorem 1.1], are only available for 𝑘 a little bit smaller than (log log 𝑥) 0.5 , which is not quite good enough 6 in our situation. To get around this, we will reformulate the above estimate using a version of the random sieve model from [1], allowing one to exploit cancellations between the different values of 𝑘.…”

“…Conjecture 1.3 (Quantitative Hardy-Littlewood prime tuples conjecture). [11,Conjecture 1.3] There exist two absolute constants 𝜀 > 0 and 𝐶 > 0 such that for all 𝑥 ≥ 10, all 𝑘 ≤ (log log 𝑥) 5 , and all tuples  = {ℎ 1 , … , ℎ 𝑘 } ⊂ [0, log 2 𝑥] of distinct integers ℎ 1 , … , ℎ 𝑘 , one has…”

“…It is common to restrict (1.1) to the admissible case 𝔖() > 0, but this is not necessary since (1.1) is very easy to establish in the non-admissible case 𝔖() = 0. In [11], 𝑘 was restricted to the slightly smaller range 𝑘 ≤ (log log 𝑥) 3 , but we will need to increase the exponent 3 to 5 for technical reasons. We remark that Conjecture 1.3 has been verified almost surely (even with this enlarged range of parameters) for a certain random model of the primes; see [1,Theorem 1.3].…”

“…It is tempting to then control the contribution of each 𝑘 separately and treat the errors by the triangle inequality, in the spirit of a well known calculation of Gallagher [6] which predicts Poissonian statistics for the number of primes in (𝑥, 𝑥 + 𝜆 log 𝑥] (which implies in particular that the mean value of (−1) 𝜋(𝑥+𝜆 log 𝑥)−𝜋(𝑥) converges to 𝑒 −2𝜆 in the limit 𝑥 → ∞ for 𝜆 fixed). Unfortunately, the range of 𝑘 needed is a little bit too large to use the strongest available estimates (due to V. Kuperberg [11]) for mean values of the singular series in this fashion; we need 𝑘 to be as large as (log log 𝑥) 4.5 or so, but quantitative estimates only currently have good error terms in the regime 𝑘 ≤ (log log 𝑥) 0.5 . However, we can proceed instead by expressing these sums (up to acceptable errors) using the random sieve model  𝑧 introduced in [1], effectively replacing the primes by this model.…”