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

Authors: Carter Bays and Richard H. Hudson
Journal: Math. Comp. 69 (2000), 861-866
MSC (1991): Primary 11A15, 11M26, 11Y11, 11Y35
Published electronically: March 10, 1999
MathSciNet review: 1651741
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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 $1.39822 \times 10^{316}$.

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Additional Information

Carter Bays
Affiliation: Department of Computer Science, University of South Carolina, Columbia, South Carolina 29208

Richard H. Hudson
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Received by editor(s): March 17, 1997
Received by editor(s) in revised form: April 1, 1998, and July 6, 1998
Published electronically: March 10, 1999
Article copyright: © Copyright 2000 American Mathematical Society