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Thomas Hagedorn has a short survey on results related to the Jacobsthal function, as well as recent computations for a_i being the first t primes for t up to 50 . It is at http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf . In his section 1, Hagedorn cites a result of Iwaniec which gives an asymptotic upper bound of order O(t log(t))^2, and he cites a more explicit upper bound that was given by Stevens as 2t^(2 + 2elog(t)). (He also cites a lower bound by Pintz which is a mild improvement on the Erdos lower bound.) I am working on replacing the bound in Stevens' result by something asymptotically smaller (involving log(log(tlog(t))). I will post it as an answer to http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update when I am confident it is valid.

UPDATE 2011.02.25 I have posted an improvement of Stevens's result as an answer to the linked question above. I welcome a review of it.END UPDATE 2011.02.25

Gerhard "Ask Me About System Design" Paseman, 2011.02.13
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Thomas Hagedorn has a short survey on results related to the Jacobsthal function, as well as recent computations for a_i being the first t primes for t up to 50 . It is at http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf . In his section 1, Hagedorn cites a result of Iwaniec which gives an asymptotic upper bound of order O(t log(t))^2, and he cites a more explicit upper bound that was given by Stevens as 2t^(2 + 2elog(t)). (He also cites a lower bound by Pintz which is a mild improvement on the Erdos lower bound.) I am working on replacing the bound in Stevens' result by something asymptotically smaller (involving log(log(tlog(t))). I will post it as an answer to http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update when I am confident it is valid.

Gerhard "Ask Me About System Design" Paseman, 2011.02.13
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Thomas Hagedorn has a short survey on results related to the Jacobsthal function, as well as recent computations for a_i being the first t primes for t up to 50 . It is at http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf . In his section 1, Hagedorn cites a result of Iwaniec which gives an asymptotic bound of order O(t log(t))^2, and he cites a more explicit bound that was given by Stevens as 2t^(2 + 2elog(t)). I am working on replacing the bound in Stevens' result by something asymptotically smaller (involving log(log(tlog(t))). I will post it as an answer to http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update when I am confident it is valid.

Gerhard "Ask Me About System Design" Paseman, 2011.02.13