The 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.)

]]>Hello 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?

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

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 ]]>

Your 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.

]]>Thank you for your prompt and helpful comment!

]]>I 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.

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