Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A refined version of the Lang-Trotter Conjecture   Let $E$ be an elliptic curve defined over the rational numbers and $r$ a
fixed integer. Using a probabilistic model consistent with the Chebotarev
theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and
Trotter conjectured an asymptotic formula for the number of primes up to $x$
which have Frobenius trace equal to $r$, where $r$ is a {\it fixed} integer.
However, as shown in this note, this asymptotic estimate cannot hold for {\it
all} $r$ in the interval $|r|\le 2\sqrt{x}$ with a uniform bound for the error
term, because an estimate of this kind would contradict the Chebotarev density
theorem as well as the Sato-Tate conjecture.
  The purpose of this note is to refine the Lang-Trotter conjecture, by taking
into account the "semicircular law", to an asymptotic formula that
conjecturally holds for arbitrary integers $r$ in the interval $|r|\le
2\sqrt{x}$, with a uniform error term. We demonstrate consistency of our
refinement with the Chebotarev theorem for a fixed division field, and with the
Sato-Tate conjecture. We also present numerical evidence for the refined
conjecture.

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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
0
The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin