Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges

Matomäki, Kaisa; Radziwiłł, Maksym; Tao, Terence

doi:10.1007/s00208-018-01801-4

Cited by 19 publications

(17 citation statements)

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“…This is the Dirichlet character analogue of [26,Lemma 4], which follows from [14,Lemma 7.1]. The proof is contained in the proof of [20,Lemma 5.2].…”

Section: Decomposing Dirichlet Polynomialsmentioning

confidence: 78%

Correlations of Almost Primes

Evans¹

2021

Preprint

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Let p 1 , . . . , p 5 be primes with p 1 p 2 , p 5 ∈ (X, 2X], let h be a non-zero integer and A > 3. Supposing that p 1 , p 3 lie in a certain range depending on X, we prove an asymptotic for the weighted number of solutions to p 3 p 4 = p 1 p 2 + h which holds for almost all 0Using the same methods and supposing instead that p 1 p 2 , p 3 p 4 have the typical factorisation we prove an asymptotic which holds for almost all 0 < |h| ≤ H with exp (log X) 1−o(1) ≤ H ≤ X log −A X and an asymptotic for the weighted number of solutions to p 3 p 4 = p 5 + h which holds for almost all 0 < |h| ≤ H with X 1/6+ε ≤ H ≤ X log −B X and B > 5.

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“…This is the Dirichlet character analogue of [26,Lemma 4], which follows from [14,Lemma 7.1]. The proof is contained in the proof of [20,Lemma 5.2].…”

Section: Decomposing Dirichlet Polynomialsmentioning

confidence: 78%

Correlations of Almost Primes

Evans¹

2021

Preprint

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“…Remark: If the loss incurred by l 2 bounds cannot be compensated by θ, one may still reach a nontrivial result. To do so, one needs to modify the approach of the proposition to make use of large value estimates, see condition (iii) of Lemma 3.2 [MRT19b].…”

Section: The Circle Methods With Minorants In Short Intervalsmentioning

confidence: 99%

Goldbach Numbers in Short Intervals -- A Nonnegative Model Approach

Lasse¹

2019

Preprint

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We decrease the length of the shortest interval for which almost all even integers in it are the sum of two primes. This is achieved by applying a version of the Circle Method that uses two minorants together with a nonnegative model for one of them. Compared to Harman's previous strongest result of this type, we in this way do not need any vector sieve type inequality and so require neither additional majorants nor strong density requirements for the minorants.

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“…We mention that results are known for affine correlations whose linear parts are pairwise independent [Mat12,Mat16], or when there is an additional, long enough average over the shift [Mik92,MRT17a,MRT17b]. See also [ABSR15,BSF17] for a function field analogue in the large q limit.…”

Section: Introductionmentioning

confidence: 99%