- One thing I have often thought about is trying to build a quasi-alternating series out of the reciprocals of the primes, so that:

- the reciprocal of every prime of the form 4n+1 is a positive term

- the reciprocal of every prime of the form 4n+3 is a negative term

- the reciprocal of 2 is a positive term, both because it is like primes of the form 4n+1 in being expressible as two squares and because such a result I imagine as necessary to give a positive sum to the whole series

The first fifty terms of the series, to give you the idea, would be:

(1/2)-(1/3)+(1/5)-(1/7)-(1/11)+(1/13)+(1/17)-(1/19)-(1/23)+(1/29)-(1/31)+(1/37)+(1/41)-(1/43)-(1/47)+(1/53)-(1/59)+(1/61)-(1/67)-(1/71)+(1/73)-(1/79)-(1/83)+(1/89)+(1/97)+(1/101)-(1/103)-(1/107)+(1/109)+(1/113)-(1/127)-(1/131)+(1/137)-(1/139)+(1/149)-(1/151)+(1/157)-(1/163)-(1/167)+(1/173)-(1/179)+(1/181)-(1/191)+(1/193)+(1/197)-(1/199)-(1/211)-(1/223)-(1/227)+(1/229)

I have summed this series using a hand calculator as far as 2500 (the first 365 primes) and found that it appears to ultimately converge to some number between 0.165 and 0.170, though sequences of positive and negative terms can be found if you look above at the early terms and make the ultimate sum hard to estimate without software which I cannot access.

By analogy with the alternating harmonic series and the fact that primes of form 4n+1 and 4n+3 will be equal in number, I have little doubt this series must (conditionally) converge. I have long wondered if there is a formula for the sum. I have often imagined the sum in terms of ln(lnk) but realise this is logically unlikely.

Can anyone confirm that this quasi-alternating series really does converge and provide some information about the sum? - --- In primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> how to obtain 70 good digits, using no odd prime at all

Off-list, Maximilian Hasler asked, amusingly:

> Can this method also be used to produce reliable

In fact, polling the odd primes really helps:

> statistics without using any data?

> If you can generalize your method, there will be

> no more polls needed at all...

> Do you realize that you are working towards a

> democracy where the government will know what the

> people wants even without them having to vote?

I found 10,000 good digits in 7 minutes, by consulting

the 24 odd primes p < 100. The result is in

http://physics.open.ac.uk/~dbroadhu/cert/cohenbig.out

David (campaign for a discriminating democracy)