[seqfan] Re: Call for more digits of 1/(3*5) + 1/(5*7) + 1/(7*11)

[Harvey P. Dale]

	This is what I computed from the first 10 million odd-prime pairs:

1,3,4,4,2,6,5,0,9,4,1,2,5,1,6,6,4,1,5,3,7,6,9,2,6,9,6,0,9,6,8,2,4,5,2,7,0,4,5,1,5,6,5,7,0,4,7,6,8,0

I'd be pretty confident the latter digits are inaccurate, but perhaps the first 10 or 11 digits???

	Best,

	Harvey

P.S. Here's the Mma program: RealDigits[Total[1/Times@@# &/@Partition[Prime[Range[2,10000000]],2,1]],10,50][[1]]
 

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of mathar
Sent: Tuesday, January 22, 2013 8:36 AM
To: seqfan at seqfan.eu
Subject: [seqfan] Call for more digits of 1/(3*5) + 1/(5*7) + 1/(7*11)

This is some call to compute additional digits to the sum 1/(3*5) +1/(5*7) + 1/(7*11)+.. in http://oeis.org/A209329 , where the denominators walk through products of adjacent odd primes.

The usual Euler-zeta-methods do not provide much guidance, because the selection of the two terms in the denominators is not just irregular with respect to the lower prime but also irregular with respect to the gap up to the next prime.

On a term-by-term basis, the partial sums of A185380 are upper estimates of the partial sums of this sum, if we look at 1/(p*(p+2)) as using an estimator p+2 as a substitute for the next prime after p.
(At the same time, convergence of A185380 proofs that A209329 converges, and A185380 is itself majored by the sum over 1/p^2, the prime zeta function at 2, A085548, which is also known to converge.) Similarly, the partial sums of sum_{odd primes q} 1/(q*(q-2)) = 0.4635423529706636936146055639434089111278372208711.. provide a lower estimate if we consider q-2 as an estimator for the prime previous to q (and remove the initial 1/3).
Accurate upper and lower limits A209329 can therefore be constructed by keeping track of the three partial sums of 1/(q*(q-2)),
1/(prime(j)*prime(j+1)) and 1/(p*(p+2)) and using the known distances of the partial sums of the two auxiliary sequence to their limits as upper and lower limits of the remaining distance of this sequence to its limit. 

So that summarizes what I actually know about that sum.

(Side node: A124012 is even more irregular and converging worse; finding useful upper and lower estimators for the terms looks much more challenging
there...)

Richard

_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/