The sum of the inverse squares of the numbers between successive twin primes (1/4^2 + 1/6^2...etc.) converges to something like 0.1029...A reference like the Online Enc. of Integer Sequences contains many in this ballpark and I see nothing self-explanatory. Can someone point me to the correct constant and the appropriate formulaic description? Thanks. DT
On Jul 24, 11:48 am, daniel tisdale <daniel6...@gmail.com> wrote:
> The sum of the inverse squares of the numbers > between successive twin primes (1/4^2 + 1/6^2...etc.) > converges to something like 0.1029...A reference like > the Online Enc. of Integer Sequences contains many > in this ballpark and I see nothing self-explanatory. > Can someone point me to the correct constant and > the appropriate formulaic description? Thanks. DT
There's no reason to think there is a simple closed form for this constant, is there? We can't even prove there are infinitely many pair of twin primes, so we don't even know for sure that yours is an infinite series. -- GM
I'm not looking for a closed form, I'm asking about the table. The sum converges whether there are infinitely many twins or not. The notation of the two major tables is a little daunting. It involves abbreviations and estimates I'm not familiar with. If someone knew which item it was, I could hunt down the details.
The table includes infinite sums or sums of many terms. It does not consist of only closed forms. See, e.g., Brun's constant.
On Jul 24, 8:19 pm, daniel tisdale <daniel6...@gmail.com> wrote:
> I'm not looking for a closed form, I'm asking about the table. > The sum converges whether there are infinitely many twins > or not. The notation of the two major tables is a little daunting. > It involves abbreviations and estimates I'm not familiar with. > If someone knew which item it was, I could hunt down the details.
> The table includes infinite sums or sums of many terms. > It does not consist of only closed forms. See, e.g., Brun's constant.
You've lost me. I don't know what you mean by "the table." I don't know what you mean by "the two major tables."
Do you mean that you have entered the sequence 1, 0, 2, 9 in the OEIS, and it has given you several sequences that start that way, and you don't know which (if any) is the sum of the inverse squares of the numbers between twin primes? -- GM
In essence, yes. Even entering seven digits, assuming I can rely on those I've calculated, produces dozens of choices. Here's a typical entry (Plouffe's remarkable site):
I don't think it's what I want, but not knowing what the function a(i) refers to, I'm not certain. The sum I'm interested in is an obvious one to calculate in studying primes. There are at least two large collections of mathematical constants. It hardly matters which one you use if you can identify the number.
Does this clarify the problem? And if the sum turns out to be finite, then I guess the "closed form" would be the somewhat unwieldy but finite sum...DT
In article <9734229.1216930034604.JavaMail.jaka...@nitrogen.mathforum.org>, daniel tisdale <daniel6...@gmail.com> wrote:
> In essence, yes. Even entering seven digits, assuming I can rely on those > I've calculated, produces dozens of choices. Here's a typical entry > (Plouffe's remarkable site):
> I don't think it's what I want, but not knowing what the function a(i) refers > to, I'm not certain.
I would interpret that notation to mean a_1/p_1 + a_2/p_2 + a_3/p_3 + ... where p_1, p_2, p_3, ... = 2, 3, 5, ... is the primes and a_1, a_2, a_3, ... = -1, -1, -1, 1, 1, ... as given, although I'm not sure whether that's a finite sum with 13 non-zero terms or an infinite sum with the numerator having period 15. I would hope there would be an explanation of notation somewhere on the site. I'm a little worried, though, because my interpretation surely leads to a negative number, while you've been asking about a positive.
> The sum I'm interested in is an obvious one to calculate > in studying primes. There are at least two large collections of mathematical > constants. It hardly matters which one you use if you can identify the > number.
> Does this clarify the problem? And if the sum turns out to be finite, then I > guess the "closed form" would be the somewhat unwieldy but finite sum...DT
Steve Finch is another guy interested in mathematical constants, in fact he has written a book about them. Maybe he knows something about this one.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
> > I don't think it's what I want, but not knowing what the function a(i) > > refers > > to, I'm not certain.
> I would interpret that notation to mean > a_1/p_1 + a_2/p_2 + a_3/p_3 + ... > where p_1, p_2, p_3, ... = 2, 3, 5, ... is the primes > and a_1, a_2, a_3, ... = -1, -1, -1, 1, 1, ... as given, > although I'm not sure whether that's a finite sum with 13 non-zero terms > or an infinite sum with the numerator having period 15. I would hope > there would be an explanation of notation somewhere on the site. > I'm a little worried, though, because my interpretation surely leads to > a negative number, while you've been asking about a positive.
That infinite sum would diverge, no? -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
> > > I don't think it's what I want, but not knowing what the function a(i) > > > refers > > > to, I'm not certain.
> > I would interpret that notation to mean > > a_1/p_1 + a_2/p_2 + a_3/p_3 + ... > > where p_1, p_2, p_3, ... = 2, 3, 5, ... is the primes > > and a_1, a_2, a_3, ... = -1, -1, -1, 1, 1, ... as given, > > although I'm not sure whether that's a finite sum with 13 non-zero terms > > or an infinite sum with the numerator having period 15. I would hope > > there would be an explanation of notation somewhere on the site. > > I'm a little worried, though, because my interpretation surely leads to > > a negative number, while you've been asking about a positive.
> That infinite sum would diverge, no?
I would expect it to, yes.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
