Hardy–Littlewood Tuple Conjecture Over Large Finite Fields

Shifted convolution and the Titchmarsh divisor problem over 𝔽q[t]

“…This plays a role in the proof, where we use that a certain Galois group is S s n [17], and we derive the statistic from an explicit Chebotarev theorem. Since we have not found the exact formulation that we need in the literature, we provide a proof in the appendix.…”

“…In [17], it is assumed that q is odd, but using [18] that restriction can now be removed for n > 2. This, in particular, implies that L ∩ F = F q (since the image of the restriction map Gal(…”

The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

“…. , a 1 , t, and, by definition of B(j), they have degree d(n) as polynomials in t. Hence by lemma 3.4, Res(δ f (t), δ f +α j (t)) has total degree 2 in the coefficients a n−1 , . .…”

“…, which is equal to (n − 1)(3n − 5) for odd n and 3n 2 − 10n + 6 for even n. In either case, we may round this up to 3n 2 …”

“…Bary-Soroker [2] uses a result similar to a part of the proof, named square independence, and computes a certain Galois group to be S r n . This computation then implies many equidistribution and independence results, proving function field analogues, in the limit of a large base field, to myriad classical problems, such as the Hardy-Littlewood conjecture, and the additive and Titchmarsh divisor problems; see Andrade et al [3] for more details and examples.…”