bio  website  icecreambreakfast.com/… 

location  Pittsburgh, Pa  
age  
visits  member for  4 years 
seen  6 hours ago  
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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.
7h

awarded  Popular Question 
Dec 21 
comment 
Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?
Does this prove, absolutely, that there can be no such $f_{q,z}(n)$, specifically for $q \in \mathbb{N}$? Or just that the usual methods can't be used to find such a function, and that it won't resemble the tidy definition for $d_z(n)$? 
Dec 20 
asked  Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$? 
Sep 24 
awarded  Autobiographer 
Jul 2 
awarded  Curious 
Apr 22 
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 
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 