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Bounded Gaps Between Primes in Chebotarev Sets
A new and exciting breakthrough due to Maynard establishes that there exist
infinitely many pairs of distinct primes with as
a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his
general method to the setting of Chebotarev sets of primes. We study
applications of these bounded gaps with an emphasis on ranks of prime quadratic
twists of elliptic curves over , congruence properties of the
Fourier coefficients of normalized Hecke eigenforms, and representations of
primes by binary quadratic forms.Comment: 15 pages. Referee comments implemented. Research in the Mathematical
Sciences 2014, 1:
The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions
Let be a
newform with squarefree level that does not have complex multiplication.
For a prime , define to be the angle for which
. Let be a closed
subinterval, and let be the
Sato-Tate measure of . Assuming that the symmetric power -functions of
satisfy certain analytic properties (all of which follow from Langlands
functoriality and the Generalized Riemann Hypothesis), we prove that if is
sufficiently large, then with an implied constant of . By letting be a short interval
centered at and counting the primes using a smooth cutoff, we
compute a lower bound for the density of positive integers for which
. In particular, if is the Ramanujan tau function, then under
the aforementioned hypotheses, we prove that
We also discuss the connection between the density of positive integers for
which and the number of representations of by certain
positive-definite, integer-valued quadratic forms.Comment: 29 pages. Significant revisions, including improvements in Theorems
1.2, 1.3, and 1.5 and a more detailed account of the contour integration, are
included. Acknowledgements are update
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