Averaging effects on irregularities in the distribution of primes in arithmetic progressions

Abstract:

Let $t$ be an integer taking on values between 1 and $x$ ($x$ real), let ${\pi _{b,c}}(t)$ denote the number of positive primes $\leqslant t$ which are $\equiv c$ $\pmod b$, and let $\operatorname {li} t$ denote the usual integral logarithm of $t$. Further, let the ratio of quadratic nonresidues of $b > 2$ to quadratic residues of $b$ be $\gamma (b)$ to 1, and let \[ A_b(x) = (1/\gamma (b)) \frac {1}{x} \left ( \sum _{\substack {t = 1\\ 1 \leqslant c \leqslant b - 1}}^x \pi _{b,c}(t) - \gamma (b) \sum _{\substack {t = 1\\1 \leqslant c’\leqslant b - 1}}^x \pi _{b,c’}(t) \right ) \] where $c$ runs over quadratic nonresidues and $c’$ runs over quadratic residues of $b$. Nearly periodic oscillations of $A_6(x) = (1/x)\Sigma _{t = 1}^x(\pi _{6,5}(t) - \pi _{6,1}(t))$ about $h(x) = (1/x)\Sigma _{t = 1}^x \operatorname {li}({t^{1/2}})/2$ are depicted in Figures 2, 3, 4 over the range of integers less than $2.5 \times {10^{11}}$. Over this range, $h(x)$ is a far better "axis of symmetry" for these oscillations than $s(x) = (1/x)\Sigma _{t = 1}^x{t^{1/2}}/\log t$ (suggested by Shanks [29]). On the other hand, recent work of W. J. Ellison [9], three letters from Andrzej Schinzel to the author, and my own considerations (see Section 4) lead to the following. In contradiction to a conjecture of Shanks [29], \[ \frac {1}{x} \left ( \sum \limits _{t = 1}^x (\pi _{6,5}(t) - \pi _{6,1}(t)) / (t^{1/2}/\log t) \right ) \nrightarrow 1\quad \text {as $x \to \infty $} . \] Moreover, I prove in Theorem 4.1 that ${A_6}(x)/h(x) \nrightarrow 1$ as $x \to \infty$, and Schinzel has provided a heuristic argument that no amount of averaging of ${A_6}(x)$ will provide an asymptotic relationship of this sort. However, let ${h^{(1)}}(x) = h(x)$, $A_6^{(1)}(x) = {A_6}(x)$, and for $k > 1$ let \[ {h^{(k + 1)}}(x) = \frac {1}{x}\sum \limits _{t = 1}^x {{h^{(k)}}(t),\quad A_6^{(k + 1)}(x) = \frac {1}{x}} \sum \limits _{t = 1}^x {A_6^{(k)}(t).} \] Assuming the truth of the generalized Riemann hypothesis for $L(s,\chi )$, $\chi$ the nonprincipal character $\bmod \,6$, we prove \[ \lim \limits _{k \to \infty } \;\overline {\lim \limits _{x \to \infty } } \;\frac {{A_6^{(k)}(x)}}{{{h^{(k)}}(x)}} = 1 = \lim \limits _{k \to \infty } \;\underline {\lim } \limits _{x \to \infty } \;\frac {{A_6^{(k)}(x)}}{{{h^{(k)}}(x)}}.\] The behavior of ${A_6}(x)$ is a special case of a far more general phenomenon. In Section 3, reasons are given why ${A_b}(x)$ can be expected to oscillate more or less symmetrically about $h(x)$ for every modulus $b > 2$.