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Old 2019-04-06, 17:38   #12
jrsousa2
 
Dec 2018
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Quote:
Originally Posted by 10metreh View Post
The expression is equal to 1 for all n (including pseudoprimes and composites).

First we deal with the finite sum. It turns out it can be written in a much simpler form.................
Wow, thanks a lot. I didn't even try that one, I've actually simplified a few other series that make up the final formula, but that one had intimidated me.

You are so good, would you mind if I propose a few other series that I'm trying to simplify? I posted them on Math Stack but those guys are often ruder than smarter.

Last fiddled with by jrsousa2 on 2019-04-06 at 17:58
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Old 2019-04-06, 17:49   #13
jrsousa2
 
Dec 2018
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Ok, so now that we know that aforementioned series is always 1, for integer x, we have that \pi^*(x) (primes and pseudo-primes) counting function is given by:

The Tex here sucks. I have no idea why the below is failing.

\pi^*(x)=2+\sum _{k=1}^{\infty } (-1)^k(2\pi )^{2k}(2k)!\sum _{j=1}^k \frac{B_{2j}}{(2k+1-2j)!(2j)!}\sum _{p=1}^{2k} \frac{(-2)^{2k-p}}{p!(2k-p)!}\sum_{n=1}^x \frac{2^{p n}}{n^{2j}}

This is not the series I'm trying to simplify yet. I have 4 convoluted series, but with the Tex here being so bad I will struggle to post them here.
I will post a link to my post on Math Stack instead.

Series that I need to simplify

Last fiddled with by Uncwilly on 2019-04-06 at 18:54 Reason: fixed tex tag
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Old 2019-04-06, 19:32   #14
10metreh
 
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There are lots of very smart people on stackexchange but they don't like being given long messy expressions like yours, with no explanation of how they were derived, and asked to simplify them. Most of them are quite busy themselves (as am I) and don't want to be drawn down a rabbit hole of calculations with no apparent end. How would they know that they're not just doing the same calculations that you did to obtain the expression, but backwards? Why should they even believe your formula for sums of squares of reciprocals of primes and pseudoprimes is correct?

Your expressions include things like the zeta function and Bernoulli numbers, which could either have appeared because you've done some deep maths or because you've looked up an expression for some functions/series and maybe don't really understand what's going on. People on stack may (rightly or wrongly) assume that anyone who actually understands that sort of maths will also be able to easily simplify an expression like the one you posted here. So if it turns out anything similar can be done to the expressions you posted there, they will conclude (rightly or wrongly) that either you're trying to do maths that's way too advanced for you, or you're just lazy.

You're more likely to get help if you show where your series come from and say what ideas you've tried that haven't worked. I only looked at the one you posted here because a) there were some obvious things to do with it, like try to get a binomial expansion to appear, and b) your computations suggested it always equalled 1, meaning that there probably was some simplification possible!

Last fiddled with by 10metreh on 2019-04-06 at 19:32
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Old 2019-04-06, 20:51   #15
jrsousa2
 
Dec 2018
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Quote:
Originally Posted by 10metreh View Post
There are lots of very smart people on stackexchange but they don't like being given long messy expressions like yours, with no explanation of how they were derived, and asked to si.....!
Messy? Excuse me?? It's not just cause you solved the problem I posted here that suddenly you're entitled to call my expressions messy, you a******.

I beg to differ as far as being rude goes, but feel free to disagree, there are a lot of a******s there and it's impossible that it was just my personal and biased opinion, it's systematic, and math-ers actually have a penchant for being blunt and rude, often unnecessarily. As for being smart, of course, not everyone in there is, but many if not most are.

The formulae you see in there are correct and I created them, I don't expect people to vet them, I verify them until I'm sure they are correct. We all make one mistake here and there, but I know that the things I do are almost always correct. They are derived from papers I released, and am about to release a 5th one.

I only wanted help to simplify the sums, there are 3 and 4 indexes at times. I doubt anybody will be able to simplify that little monster, Lol, but I wanted to be sure, because that's probably above my head (though there's been times that I've solved things I doubted I could, sometimes we underestimate our capacity or overestimate the problem).

Last fiddled with by petrw1 on 2019-04-06 at 21:37 Reason: And another expurgated
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Old 2019-04-07, 01:52   #16
CRGreathouse
 
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Quote:
Originally Posted by jrsousa2 View Post
Messy? Excuse me?? It's not just cause you solved the problem I posted here that suddenly you're entitled to call my expressions messy, you a******.
Your expression was messy. If you can't deal with someone pointing out a fairly self-evident reality, you really need to find somewhere else to get your mathematical help. We try to be a welcoming community, but you can't bite the hand that feeds you. More importantly, we need to know where things came from. This expression didn't just fall out of the sky. How did you derive it, what kind of process was used? This will help us give you meaningful feedback when you post formulas that aren't easy enough to solve in closed form.

And please be nice.
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