arXiv:1406.2658v1 [math.NT] 10 Jun 2014 On the ratio of consecutive gaps between primes J´ anos Pintz ∗ Dedicated to Helmut Maier on the occasion of his 60 th birthday 1 Introduction The difference between the consecutive primes, the expression d n = p n+1 − p n , (1.1) where P = {p i } ∞ i=1 denotes the set of primes, has been investigated probably since the time of the Greeks. The Twin Prime Conjecture asserts d n =2 infinitely often. (1.2) This conjecture settles at the conjectural level the small values of d n . Concerning the large values even the suitable conjecture is not completely clear. However, it seems to be that Cram´ er’s conjecture [Cra1, Cra2] lim sup n→∞ d n log 2 n = C 0 =1 (1.3) is near to the truth. Granville suggested [Gra1, Gra2], just based on the famous matrix method of Helmut Maier that the correct value of C is instead of 1 slightly larger C =2e −γ > 1. (1.4) However, most mathematicians agree that the correct maximal order of d n should be (log n) 2+o(1) . * Supported by OTKA Grants NK104183, K100291 and ERC-AdG. 321104. 1