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


Author: Richard H. Hudson
Journal: Math. Comp. 44 (1985), 561-571
MSC: Primary 11N13; Secondary 11N05, 11Y35
DOI: https://doi.org/10.1090/S0025-5718-1985-0777286-7
MathSciNet review: 777286
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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 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

$\displaystyle {A_b}(x) = (1/\gamma (b))\frac{1}{x}\left( {\mathop \sum \limits_... ... {t = 1}\limits_{1 \leqslant c'\leqslant b - 1} }^x \,{\pi _{b,c'}}(t)} \right)$

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],

$\displaystyle \frac{1}{x}\left( {\sum\limits_{t = 1}^x {({\pi _{6,5}}(t) - {\pi... ...)/({t^{1/2}}/\log t)} } \right) \nrightarrow 1\quad {\text{as}}\;x \to \infty .$

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

$\displaystyle {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

$\displaystyle \mathop {\lim }\limits_{k \to \infty } \;\overline {\mathop {\lim... ...derline{\lim}} \limits_{x \to \infty } \;\frac{{A_6^{(k)}(x)}}{{{h^{(k)}}(x)}}.$

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$.


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DOI: https://doi.org/10.1090/S0025-5718-1985-0777286-7
Article copyright: © Copyright 1985 American Mathematical Society

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