Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

Abstract

We study asymptotics of sums of the form 0true∑X<n⩽2XnormalΛ(n)normalΛ(n+h), 0true∑X<n⩽2Xdk(n)dl(n+h), 0true∑X<n⩽2XnormalΛ(n)dk(n+h), and 0true∑nnormalΛ(n)normalΛ(N−n), where normalΛ is the von Mangoldt function, dk is the kth divisor function, and N,X are large. Our main result is that the expected asymptotic for the first three sums holds for almost all h∈[−H,H], provided that Xσ+ε⩽H⩽X1−ε for some ε>0, where σ:=833=0.2424⋯, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa and Baier–Browning–Marasingha–Zhao, who obtained statements of this form with σ replaced by 13. We obtain an analogous result for the fourth sum for most N in an interval of the form [X,X+H] with Xσ+ε⩽H⩽X1−ε. Our method starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean‐value estimates for certain Dirichlet polynomials associated to ‘Type d3’ and ‘Type d4’ sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type d4 sum is treated similarly using the classical L2 mean value theorem and the classical van der Corput exponential sum estimates. In a sequel to this paper we will obtain related results for the correlations involving dkfalse(nfalse) for much smaller values of H but with weaker bounds.

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

“…This paper is a sequel to our previous work [14]. As in our previous paper, we are interested in the correlations 1 X<n≤2X d k (n)d l (n + h) and…”

“…For non-integer k we define d k (n) as the nth coefficient of the Dirichlet series of ζ(s) k , where ζ(s) is the Riemann zeta-function. In contrast to [14], we omit the case of Λ(n)Λ(n + h) from consideration as we will have nothing new to say for these correlations. When h = 0 is fixed, X goes to infinity, and k ≥ l ≥ 2 are fixed integers, there are well-established conjectures for the asymptotic values of expressions in (1).…”

“…This is also wide open as soon as k, l ≥ 3. See [14] for further discussion of these conjecture as well as previous progress on them . We expect the asymptotic estimates (2) and (3) to remain true (though possibly with a worse error term) when k and l are not integers. In that setting P k,l,h (z) and Q k,h (z) are no longer polynomials, but rather transcendental functions that can be expanded into asymptotic series of decreasing powers of z, and with the highest term a constant multiple of z k+l−2 and z k−1 respectively.…”

Discorrelation Between Primes in Short Intervals and Polynomial Phases

“…Recalling the special shape of a (j) r , van der Corput exponential sum estimates and the zero-free region for L(s, χ) imply that (2.2) holds unless |t| ≤ 2 n 0 H 0 (log N ) A+B+57 and χ is principal (this follows directly e.g. from [15,Lemma 2.7]). Lemma 2.3 is thus applicable if |t| > 2 n 0 H 0 (log N ) A+B+57 .…”

Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions

“…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.…”