Zeroes of Dirichlet $L$-functions and irregularities in the distribution of primes

Abstract:

Seven widely spaced regions of integers with $\pi _{4,3} (x) <\pi _{4,1} (x)$ have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of $\pi _{4,3}(x) -\pi _{4,1}(x)$ including the sign change (axis crossing) regions, in time linear in $x,$ using zeroes of $L(s,\chi ), \chi$ the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to $10^{1000}$. Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference $\text {li} (x)- \pi (x)$ with fine resolution out to and beyond the region in the vicinity of $6.658 \times 10^{370}$ discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of $1.39822 \times 10^{316}$.