Bounds and conjectures for additive divisor sums

Ng, Nathan; Thom, Mark

doi:10.7169/facm/1735

Cited by 9 publications

(17 citation statements)

References 38 publications

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“…Using the bounds (68), we can argue as before to show that But as p k,q , p l,q are polynomials in log X of degree at most k − 1, l − 1 respectively, this follows from direct calculation (using (68) to justify the convergence of the summation). An explicit formula for the polynomial P k,l,h may be computed by using the calculations in [9], but we will not do so here.…”

Section: By the Fourier Inversion Formula We Havementioning

confidence: 99%

“…Now we turn to the divisor correlation conjecture . These correlations have been studied by many authors . When

k = 2

, the conjecture is known to be true with a somewhat worse error term.…”

Section: Introductionmentioning

confidence: 99%

“…where n is a random variable drawn from the profinite integersẐ with uniform Haar probability measure, d k,p (n) := vp(n)+k−1 k−1 is the local component of d k at p (with the p-valuation v p (n) being the supremum of all j such that p j divides n), and Λ p (n) := p p−1 1 p∤n is the local component of Λ. See [68] for an explanation of these heuristics and a verification of the asymptotic (11), as well as an explicit formula for the local factor S k,l,p (h). For comparison, it is easy to see that…”

Section: Introductionmentioning

confidence: 99%

“…In the cases

l ⩾ 3

, a power savings

0true \sum_{X < n ⩽ 2 X} d_{2} (n) d_{l} (n + h) = P_{2, l, h} (log X) X + O (X^{1 - δ_{l} + o false(1 false)})

for exponents

δ_{l} > 0

, is known . See for further references and surveys of the problem. Finally, we remark that a function field analogue of has been established in , but with an error term that is only bounded by

O_{k, l} false(q^{- 1 / 2} false)

times the main term (so the result pertains to the ‘large

q

limit’ rather than the ‘large

n

limit').…”

Section: Introductionmentioning

confidence: 99%

“…When

k, l ⩾ 3

, no unconditional proof of even the weaker asymptotic

0true \sum_{X < n ⩽ 2 X} d_{k} (n) d_{l} (n + h) = P_{k, l, h} (log X) X + o (X {prefixlog}^{k + l - 2} X)

is known. However, upper and lower bounds of the correct order of magnitude are available; see for example, .…”

Section: Introductionmentioning

confidence: 99%

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

Matomäki

Radziwiłł

Tao

2018

Proc. London Math. Soc.

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We study asymptotics of sums of the form 0true∑X0, 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.

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Section: By the Fourier Inversion Formula We Havementioning

confidence: 99%

“…Now we turn to the divisor correlation conjecture . These correlations have been studied by many authors . When

k = 2

, the conjecture is known to be true with a somewhat worse error term.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In the cases

l ⩾ 3

, a power savings

0true \sum_{X < n ⩽ 2 X} d_{2} (n) d_{l} (n + h) = P_{2, l, h} (log X) X + O (X^{1 - δ_{l} + o false(1 false)})

for exponents

δ_{l} > 0

, is known . See for further references and surveys of the problem. Finally, we remark that a function field analogue of has been established in , but with an error term that is only bounded by

O_{k, l} false(q^{- 1 / 2} false)

times the main term (so the result pertains to the ‘large

q

limit’ rather than the ‘large

n

limit').…”

Section: Introductionmentioning

confidence: 99%

“…When

k, l ⩾ 3

, no unconditional proof of even the weaker asymptotic

0true \sum_{X < n ⩽ 2 X} d_{k} (n) d_{l} (n + h) = P_{k, l, h} (log X) X + o (X {prefixlog}^{k + l - 2} X)

is known. However, upper and lower bounds of the correct order of magnitude are available; see for example, .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Matomäki

Radziwiłł

Tao

2018

Proc. London Math. Soc.

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Mean values of long Dirichlet polynomials with higher divisor coefficients

Hamieh

2022

Advances in Mathematics

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Higher uniformity of arithmetic functions in short intervals I. All intervals

Matomäki,

Shao,

Tao

et al. 2023

Forum of Mathematics, Pi

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We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

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Bounds and conjectures for additive divisor sums

Cited by 9 publications

References 38 publications

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

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

Mean values of long Dirichlet polynomials with higher divisor coefficients

Higher uniformity of arithmetic functions in short intervals I. All intervals

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