bio | website | icecreambreakfast.com/… |
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location | Pittsburgh, Pa | |
age | ||
visits | member for | 3 years, 7 months |
seen | 6 hours ago | |
stats | profile views | 863 |
I'm highly interested in algorithms for computing arithmetic summatory functions, Linnik's identity, arithmetic summatory functions generally, and especially the role of divisor-style functions in those spaces.
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 |
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. |