The General Additive Divisor Problem and Moments of the Zeta-Function

On the general additive divisor problem

“…Thus Q k (x, q) is a polynomial of degree 2k − 2 in x whose coefficients depend on q and may be explicitly evaluated. The work of Conrey and Gonek [2] predicts, as stated in (2.2), that D k (N, h) is well approximated by 2N N S k (x, h) dx, which equals N times a polynomial in log N of degree 2k − 2, all of whose coefficients depend on h and k. This is in agreement with [5] (when k = 3) and [6] (in the general case), although the shape of the polynomial in question is somewhat different. Conrey and Gonek even predict that uniformly…”

“…8]). In [6] the research begun in [5] was continued, and a plausible heuristic evaluation of the polynomial P 2k−2 (x; h) in (1.4) was made. Yet another (heuristic) evaluation of the sum in (1.5) was made later by Conrey and Gonek [2] in 2001.…”

“…Yet another (heuristic) evaluation of the sum in (1.5) was made later by Conrey and Gonek [2] in 2001. Moreover, it was shown in [6] that, for a fixed integer k ≥ 3 and any fixed ε > 0, we have…”

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

“…This conjecture appears in the work of Ivić [20] and Conrey & Gonek [5], and from their work, with some effort, we can explicitly write the conjectural leading coefficient for the desired polynomial. The conjecture over Z states that…”

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

“…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). For instance it is conjectured in [21], [11], [3,Conjecture 3] that X<n≤2X d k (n)d l (n + h) = P k,l,h (log X) · X + O ε (X 1/2+ε ) (2) and X<n≤2X Λ(n)d k (n + h) = Q k,h (log X)X + O ε (X 1/2+ε )…”