Brun theorem

on prime twins

The series $\sum 1/p$ is convergent if $p$ runs through all (the first members of all) prime twins. This means that even if the number of prime twins is infinitely large, they are still located in the natural sequence rather sparsely. This theorem was demonstrated by V. Brun [1]. The convergence of a similar series for generalized twins was proved at a later date.

References

[1]	V. Brun, "La série $\frac1{5} + \frac1{7} + \frac1{11} + \frac1{13} + \frac1{17} + \frac1{19} + \frac1{29} + \frac1{31} + \frac1{41} + \frac1{43} + \frac1{59} + \frac1{61} + \ldots$ où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie" Bull. Sci. Math. (2) , 43 (1919) pp. 100–104; 124–128
[2]	E. Trost, "Primzahlen" , Birkhäuser (1953)

References

[a1]	H. Halberstam, H.-E. Richert, "Sieve methods" , Acad. Press (1974)

Comments

The value of the sum over all elements of prime twins has been estimated as 1.9021605831….

References

[b1]	Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2