On the fluctuations of Littlewood for primes of the form $4n\not =1$

Abstract:

Let ${\pi _{b,c}}(x)$ denote the number of primes $\leqslant x$ which are $\equiv c\;\pmod b$. Among the first 950,000,000 integers there are only a few thousand integers n with ${\pi _{4,3}}(n) < {\pi _{4,1}}(n)$. The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for $L(s)$. The discovery that the majority of all Integers n less than $2 \times {10^{10}}$ with ${\pi _{4,3}}(n) < {\pi _{4,1}}(n)$ are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.