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FeatureSmall gapsMoving on to small gaps, we encounter one of the fascinatingmathematical thrillers of the last decade. Note first that theprime gaps arep n+1 − p n ≥ 2,the value 2 corresponding to the prime twins. Thus provingthe expected fact that there exist an infinity of prime twinswould be solving one of the important conjectures concerningsmall prime gaps. But evenlim inf δ(n) = 0n→∞was not known in 2002 when Goldston announced, on behalfof himself and Yildirim, an amazing result: he claimed theproof thatδ(n)/ log(n) 1/16 < 1for infinitely many primes and thus a sharpening of the vanishingof the limes inferior, a sharpening which is muchstronger than the corresponding result of Rankin for largegaps. The proof was unfortunately erroneous and was forgottenfor several years, until the same authors, together with J.Pintz, made the breakthrough in 2005, giving a correct proofoflim inf δ(n) = 0.n→∞While the three authors joined forces with Graham, consideringvarious related problems, the next major breakthroughfollowed in 2013 and was achieved by Y. Zhang, who refinedthe method of Goldson, Pintz and Yildirim, thus proving afixed upper boundlim inf (p n+1 − p n ) < 7 · 10 7 .n→∞The upper bound has since been successively improved overthe whole year and, at the time of writing, had dropped below5 · 10 5 ; see [1] for a tabular presentation of the detailedevolution of the proven bounds.The repartition of primes can be considered on a more detailedbasis by asking the question about repartition of primesin arithmetic sequences; concretely, given two coprime integersm, a, one wishes to gather information about the functionπ(x; m, a) = |{p ∈ P ≤x : p ≡ a mod m}|.Dirichlet had already established the asymptotic behaviourϕ(m) · π(x; m, a)lim= 1x→∞ π(x)but more subtle questions exist concerning, for instance, theasymptotics of the difference|π(x; m, a) − π(x; m, b)|.An important conjecture due to Elliott and Halberstam refersto these oscillations; more precisely, considering the maximaldeviationE(x; m) = maxπ(x) (a,m)=1π(x; m, a) −ϕ(m) ,Elliott and Halberstam stipulate the following.Conjecture 2. For every θ with 0 < θ < 1 and for everyA > 0 there exists a constant C > 0 such thatCxE(x; m) ≤log A (x) .1≤m≤x θThe conjecture has been proven for all θ