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Jun 13, 2019 at 8:52 comment added Jan-Christoph Schlage-Puchta A bound involving $k^2$ might be achievable, but I don't think anybody has done it yet. For the upper bound Selberg's sieve might suffice. The lower bound is probably more difficult. A good starting point might be Iwaniec's work on Jacobsthal function.
Jun 12, 2019 at 19:35 comment added Martin Hicks Thanks for your answer. What do you think about the correct order? Is it $O(k^2 \log k)$?
Jun 12, 2019 at 13:03 vote accept Martin Hicks
Jun 12, 2019 at 10:43 history answered Jan-Christoph Schlage-Puchta CC BY-SA 4.0