A note on large gaps between consecutive primes in arithmetic progressions
Let \((a,q)=1\) and let \(G(x;a,q)\) denote the maximum of \(p_{n+1}-p_ n\), taken over consecutive primes \(p_ n\), \(p_{n+1}\equiv a\pmod q\), with \(p_ n\leq x\). Then for any constant \(C>0\) it is shown that \[ G(qx;q,a)\geq(e^ \gamma+o_ C(1))\varphi(q)\log x\log_ 2{\log_ 4x\over(\log_ 3x)^ 2}, \] uniformly for \(\omega(q)\leq\{C\log_ 2x(\log_ 4x)/(\log_ 3x)\}\). Here \(\log_ n\) denotes the iterated logarithm as usual. The case \(q=1\) is the well-known result of \textit{R. A. Rankin} [Proc. Edinb. Math. Soc., II. Ser. 13, 331-332 (1963; Zbl 0121.047)]. The proof depends on a variant of a result of \textit{C. Pomerance} [J. Number Theory 12, 218-223 (1980; Zbl 0436.10020)].
scientific article; zbMATH DE number 166050
scientific article