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 | References | Similar Articles | Additional Information

Abstract: Let *t* be an integer taking on values between 1 and *x* (*x* real), let denote the number of positive primes which are , and let li *t* denote the usual integral logarithm of *t*. Further, let the ratio of quadratic nonresidues of to quadratic residues of *b* be to 1, and let

*c*runs over quadratic nonresidues and runs over quadratic residues of

*b*.

Nearly periodic oscillations of about are depicted in Figures 2, 3, 4 over the range of integers less than . Over this range, is a far better "axis of symmetry" for these oscillations than (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],

Assuming the truth of the generalized Riemann hypothesis for , the nonprincipal character , we prove

The behavior of is a special case of a far more general phenomenon. In Section 3, reasons are given why can be expected to oscillate more or less symmetrically about for every modulus .

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0777286-7

Article copyright:
© Copyright 1985
American Mathematical Society