This is a postscript to the previous blog post which was concerned with obtaining heuristic asymptotic predictions for the correlation

for the divisor function , in particular recovering the calculation of Ingham that obtained the asymptotic

when was fixed and non-zero and went to infinity. It is natural to consider the more general correlations

for fixed and non-zero , where

is the order divisor function. The sum (1) then corresponds to the case . For , , and a routine application of the Dirichlet hyperbola method (or Perron’s formula) gives the asymptotic

or more accurately

where is a certain explicit polynomial of degree with leading coefficient ; see e.g. Exercise 31 of this previous post for a discussion of the case (which is already typical). Similarly if . For more general , there is a conjecture of Conrey and Gonek which predicts that

for some polynomial of degree which is explicit but whose form is rather complicated (one has to compute residues of a various complicated products of zeta functions and local factors). This conjecture has been verified when or , by the work of Linnik, Motohashi, Fouvry-Tenenbaum, and others, but all the remaining cases when are currently open.

In principle, the calculations of the previous post should recover the predictions of Conrey and Gonek. In this post I would like to record this for the top order term:

Conjecture 1If and are fixed, thenas , where the product is over all primes , and the local factors are given by the formula

where is the degree polynomial

where

and one adopts the conventions that and for .

For instance, if then

and hence

and the above conjecture recovers the Ingham formula (2). For , we have

and so we predict

where

Similarly, if we have

and so we predict

where

and so forth.

As in the previous blog, the idea is to factorise

where the local factors are given by

(where means that divides precisely times), or in terms of the valuation of at ,

We then have the following exact local asymptotics:

Proposition 2 (Local correlations)Let be a profinite integer chosen uniformly at random, let be a profinite integer, and let . Then

(For profinite integers it is possible that and hence are infinite, but this is a probability zero event and so can be ignored.)

Conjecture 1 can then be heuristically justified from the local calculations (2) by various pseudorandomness heuristics, as discussed in the previous post.

I’ll give a short proof of the above proposition below, basically using the recursive methods of the previous post. This short proof actually took be quite a while to find; I spent several hours and a fair bit of scratch paper working out the cases laboriously by hand (with some assistance and cross-checking from Maple). Here is an unorganised sample of some of this scratch, just to show how the sausage is actually made:

It was only after expending all this effort that I realised that it would be much more efficient to compute the correlations for all values of simultaneously by using generating functions. After performing this computation, it then became apparent that there would be a direct combinatorial proof of (6) that was shorter than even the generating function proof. (I will not supply the full generating function calculations here, but will at least show them for the easier correlation (5).)

I am confident that Conjecture 1 is consistent with the explicit asymptotic in the Conrey-Gonek conjecture, but have not yet rigorously established that the leading order term in the latter is indeed identical to the expression provided above.

We now prove (5). Here we will use the generating function method. From the binomial formula we have the formal expansion

for any . By the geometric series formula, it thus suffices to show that

in the ring of formal power series in .

With probability , is coprime to and thus . In the remaining probability event, for some with the same distribution as the original random variable , and . This leads to the identity

and the claim then follows from a brief amount of high-school algebra.

Now we prove (6). By (3), our task is to show that

The above generating function method will establish this (and, as I said above, this was how I originally discovered the formula), but we give here a combinatorial proof. We first deal with the case when is not divisible by . In this case, at least one of and must equal ; we can phrase this fact algebraically as the identity

By linearity (and translation invariance) of expectation and (5) we thus have

which gives (7) in this case.

Now suppose that for some (profinite) integer . With probability , is coprime to , so that ; in particular (8) holds here. In the remaining probability event, we can write . From (4) and Pascal triangle identities, we have

and similarly

and hence on subtracting from both equations, multiplying, and rearranging we have a variant of (8):

Putting this together, we see that

and hence by (5) as before

The claim is now easily verified from an induction on , as well as many applications of Pascal triangle identities.

## 24 comments

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31 August, 2016 at 12:31 pm

sylvainjulienIn conjecture 1, I get a message ‘formula doesn’t parse’ after ‘local factors’.

[Corrected now – T.]31 August, 2016 at 5:17 pm

Dejan KovacevicIt’s interesting to see how the sausage is actually made ;-)

Not sure about the others, but I am not using paper at all, although there are no fully appropriate software tools on the market for what Terry showed. Maple and Mathematica are useful just for small subset for all that work (and one to follow)…Here is my ‘meatball shop’:

http://ateravis.com/owncloud/index.php/s/jG3A3WvFA7dw4RV

31 August, 2016 at 5:43 pm

AnonymousIs it possible to get similar conjectures for instead of (or perhaps in addition to) ?

31 August, 2016 at 6:05 pm

Terence TaoCertainly! Actually the two types of functions are related to each other by various linear identites such as , so it is fairly easy to pass from one to the other.

19 June, 2021 at 11:07 am

Irene KlaasThe identity you mentioned appears to be wrong. I get from your notation that

\[\tau=\tau_2.\]

In fact, if you take $h \rightarrow 0$ then your factor in (2) explodes. You get logarithmic divergence and this is what you would expect, isn’t it?

[See my correction from Sep 1 2016 below – T.]1 September, 2016 at 1:52 am

AnonymousSince is meromorphic on for , while for the more general generating function , it was proved (see Ramanujan’s collected papers, p. 339) by Estermann (probably in his paper “On certain functions represented by Dirichlet series” Proc. London Math. Soc.(2) 27 (1928) 435-448) that for , the function is meromorphic in the right half plane and, except the cases or , has the line as its natural boundary!

Therefore it is not clear how such linear identities representing for (whose generating function has the imaginary axis as its natural boundary) as linear combinations of (whose generating function is a polynomial in and therefore meromorphic on ) can possibly exist?

1 September, 2016 at 7:46 am

Terence TaoOops, you’re right; the identities I had in mind were only true locally, for instance . So the local factors for correlations can be derived from those of correlations, which can then be multiplied together to establish the global prediction. But a proof of the global correlation estimates for do not directly imply those for or vice versa. Nevertheless, the theory of both functions ought to be very similar. For instance agrees with on squarefree numbers.

1 September, 2016 at 4:56 am

SamI’m so happy to see your work.We’re gonna meet one day.

1 September, 2016 at 2:11 pm

AnonymousI’d be interested in a higher res picture of the scratch paper, to get a look at the actual calculations.

2 September, 2016 at 5:55 am

arch1Is there a j^2 missing from the 3rd term in the numerator of the expression for ?

6 September, 2016 at 10:17 pm

Heuristic computation of correlations of higher order divisor functions — What’s new – hakoblog[…] via Heuristic computation of correlations of higher order divisor functions — What’s new […]

10 September, 2016 at 12:41 am

Dejan KovacevicQuestion: I noticed that some sheets of paper on shared image contains either special characters or, more likely, some symbolic/graphic notation. I wonder if that is something formal/established or it is a mechanism which you use to simplify analysis?

10 September, 2016 at 11:04 pm

KenIt is already published before, isn’t? http://vixra.org/abs/1609.0030

18 September, 2016 at 9:27 am

AnonIs there a nice form known for the ordinary generating function of $\tau_k(n)$, i.e. $\sum_{n=1}^{\infty} \tau_k(n) x^n$? Or a nice way to estimate this for a fixed value of $x$ with $|x|<1$?

This came up recently in the course of some calculations I was making, and I can only seem to find info about the Dirichlet generating function of $\tau_k(n)$.

18 September, 2016 at 9:37 am

Terence TaoI doubt there is a closed form expression for . One can use the geometric series formula to remove one of the summations, e.g. , but after that there isn’t much else one can do. (The divisor functions have multiplicative structure but not much additive structure, which is the main reason why they have a nice Dirichlet series but not such a nice generating function.)

On the other hand, one can use summation by parts to estimate an expression such as in terms of partial sums . Roughly speaking, for of the form , a sum of the form should behave like (a smoothed version) of , since the range is roughly where the weight is large.

18 September, 2016 at 9:41 am

AnonThank you for your prompt and helpful comment!

18 September, 2016 at 1:24 pm

AnonymousYour example for the generating function for is a particular Lambert series, and indeed the Wikipedia article on Lambert series gives the generating function for (for any complex ) as a simple Lambert series.

11 October, 2016 at 10:40 pm

AnonymousVery enlightening!

11 October, 2016 at 10:42 pm

AnonymousThe conclusion of this paper is true?

The Distribution of Prime Numbers in an Interval

American Journal of Mathematics and Statistics, 2015 5(6), pp. 325-328

10.5923/j.ajms.20150506.01

http://article.sapub.org/10.5923.j.ajms.20150506.01.html

26 December, 2016 at 10:45 am

heverson01“The first mathematical fact that amazed me was the Pythagorean theorem,” said mathematician Daina Taimina of Cornell University (USA). “I was a child and it seemed so incredible that it worked in geometry and worked with numbers!

https://johanneshaertel.wordpress.com/2016/12/24/8-os-beneficios-do-abacate/

23 February, 2017 at 1:42 pm

Kevin SmithHello Terry, I am wondering if the asymptotic correlations will remain the same here when $k$, $l$ are positive real numbers and not necessarily integer. What is your opinion on this?

25 February, 2017 at 10:45 am

Terence TaoThe top order term in the asymptotics should be basically the same (replacing factorials with Gamma functions as appropriate, of course). The lower order terms are more subtle, I think one should get an infinite asymptotic series in powers of now, rather than just a polynomial. (One already sees this behavior just for the 1-point correlations when is non-integer.)

27 December, 2017 at 9:55 am

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges | What's new[…] discussed in this previous post, one heuristically expects an asymptotic of the […]

22 February, 2020 at 1:46 am

James MoriartyApparently you were unhappy with Windows Phone 8/8.1. What’s the feature you missed? What’s what you didn’t like?