bio  website  icecreambreakfast.com/… 

location  Pittsburgh, Pa  
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visits  member for  3 years, 3 months 
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I'm highly interested in algorithms for computing arithmetic summatory functions, Linnik's identity, arithmetic summatory functions generally, and especially the role of divisorstyle functions in those spaces.
1d

asked  Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zeta Zeros? 
Apr 1 
awarded  Notable Question 
Dec 13 
awarded  Tumbleweed 
Dec 6 
asked  In this prime counting identity, can this limit of a sum be expressed as integrals? 
Oct 23 
asked  A generalization of alternating series involving modulus? 
Jun 25 
awarded  Promoter 
Jan 14 
revised 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Added more justification and examples for the time and space bound claims. 
Dec 20 
comment 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Charles: I suppose I did mean $O(n^\epsilon)$. Anyway, in neither implementation are any values memoized. This is especially easy to see in the first C implementation I linked to. If you skim the 25 lines of code of the actual algorithm, you can see it's literally transcribing the identity I'm asking about verbatim. No tricks or cleverness (aside from specializing D_1 for constant time speed improvement). Aside from pushing functions onto the stack to a depth of $\log_2 n$, no memory is ever allocated. Also notice my edit; sorry about that. 
Dec 20 
revised 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Fixed an error in the identity  a specific specialization is required for it to work. 
Dec 20 
revised 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Fixed Memory Bound notation 
Dec 20 
revised 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Added some clarifying references 
Dec 19 
comment 
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
Oh  epsilon. Negligible. Probably $\log n$ or thereabouts, because the recursive depth of $D_{k,a}(n)$ grows at roughly $\log n$ Essentially it doesn't use any memory at all for practical purposes. 
Dec 19 
asked  Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes? 
Jun 22 
accepted  The relationship between the Dirichlet Hyperbola Method, the prime counting function, and Mertens function 
Jun 22 
comment 
The relationship between the Dirichlet Hyperbola Method, the prime counting function, and Mertens function
Thanks Terry. Given that my question was a bit vague, this is some great stuff for me to chew on. 
Jun 8 
awarded  Enthusiast 
Jun 6 
revised 
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
Clarified question further to highlight my focus on the divisor summatory function 
Jun 6 
comment 
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
That exhausts my ideas, I think. So that's the core of my "other techniques" question  1) are there other ways of expressing $H_k(n)$ (or $D_k(n)$) that lead to any interesting observations in my (4)? And particularly, are there any clever ideas for combining any of those alternative expressions with symmetries between, say, $H_2(n)$ and $H_3(n)$ and $H_4(n)$ and so on, which all jump in ways that share some connections? 
Jun 6 
comment 
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
Obviously my Extra section is another such idea for $H_k(n)$. Additionally, $D_k(n)$ and $H_k(n)$ can be expressed with generalizations of the Dirichlet Hyperbola method. If you look at mathoverflow.net/questions/97694/…, equation (4) from that page one way of writing that generalization. On that page, $D_{2,2}(n)$ is your $H_2(n)$, $D_{3,2}(n)$ is your $H_3(n)$, and so on. I'm still trying to figure out if using those equations in my (4) above leads anywhere interesting. 
Jun 6 
comment 
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
So, for example, $D_k(n)$ can be expressed as residues, as per books.google.com/books?id=jT9gjGipNDUC&pg=PA352 . (When I tried taking the principal terms for $H_k(n)$ and using them as an approximation in the left side of my (1), I was just left with the logarithmic integral). $D_2(n)$ can also be expressed with the Voronoi summation formula, as per books.google.com/books?id=jT9gjGipNDUC&pg=PA83 . I haven't been able to track down if there are comparable formulas for $D_k(n)$ (which would immediately yields expressions for $H_k(n)$). (cont...) 