On the general additive divisor problem

Ivić, Aleksandar; Wu, Jie

doi:10.1134/s0081543812010117

Cited by 5 publications

(12 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The next corollaries, applying such an intimate link, improve recent results [5,6] on an additive divisor problem for (first two conditionally, third unconditionally). They concern the deviation of ; that is,…”

Section: Theorem 2 If General Conjecture CL Holds Thensupporting

confidence: 62%

Generations of Correlation Averages

Coppola

Laporta

2014

Journal of Numbers

View full text Add to dashboard Cite

We give a general link between weighted Selberg integrals of any arithmetic functionfand averages offcorrelations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functionsdk(n), gauged by the Cesaro weight in the short intervaln∈x-H,x+Hand improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2kth moments of Riemann zeta function on the critical line are derived. We also give general requirements onfthat allow our treatment forfweighted Selberg integrals.

show abstract

Section: Theorem 2 If General Conjecture CL Holds Thensupporting

confidence: 62%

Generations of Correlation Averages

Coppola

Laporta

2014

Journal of Numbers

View full text Add to dashboard Cite

show abstract

“…We now mention several avenues of possible future research. 31 (1) Improve the lower bounds for D k,ℓ (x, h). One might attempt to use inequalities of the shape τ k (n) ≥ where r ≥ 2, k 1 , .…”

Section: Discussionmentioning

confidence: 99%

Bounds and conjectures for additive divisor sums

Ng¹,

Thom²

2019

Funct. Approx. Comment. Math.

View full text Add to dashboard Cite

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study D k,ℓ (x) = n≤x τ k (n)τ ℓ (n + h) where τ k and τ ℓ are the k-th and ℓ-th divisor functions. The main result is a lower bound of the correct order of magnitude for D k,ℓ (x, h), uniform in h. In addition, the conjectural asymptotic formula for D k,ℓ (x, h) is studied. Using an argument of Ivić [29], [30] and Conrey-Gonek [9] the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao [45].This last sum has been extensively studied. For k = 1, this sum is trivial. For k = 2, there is a rich theory connecting this sum to the spectral theory of automorphic forms. However, for k > 2, this sum is mysterious and there are few results. Nevertheless, there is the following conjecture:

show abstract

“…for large x, which is notoriously difficult if j ≥ 3, see e.g. [10], [1], and, for j = 3, [9]. In the present paper, we consider the rather less well-studied jth non-trivial divisor function c j , which counts the ordered proper factorisations of a positive integer into j factors, each of which is greater than or equal to 2.…”

Section: Introductionmentioning

confidence: 99%

Some properties and applications of non-trivial divisor functions

et al. 2019

View full text Add to dashboard Cite

The jth divisor function d j , which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known arithmetic function; in particular, d 2 (n) gives the number of divisors of n. However, the jth non-trivial divisor function c j , which counts the ordered proper factorisations of a positive integer into j factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated divisor functions c (r) j , whose definition is motivated by the sum-over divisors recurrence for d j . After reviewing properties of d j , we study analogous properties of c j and c (r) j , specifically regarding their Dirichlet series and generating functions, as well as representations in terms of binomial coefficient sums and hypergeometric series. We also express their ratios as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products in some cases. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée, and hence sum-and-distance systems of integers.

show abstract

On the general additive divisor problem

Abstract: We obtain a new upper bound for the sumis the (expected) error term in the asymptotic formula for N

Cited by 5 publications

References 17 publications

Generations of Correlation Averages

Generations of Correlation Averages

Bounds and conjectures for additive divisor sums

Some properties and applications of non-trivial divisor functions

Contact Info

Product

Resources

About