Abstract
Let f(z) = ∑∞n=1 a(n)e2πinz ∈ Snewk (Γ(N)) be a newform of even weight k ≥ 2 that does not have complex multiplication. Then a(n) ∈ ℝ for all n; so for any prime p, there exists θp ∈ [0, π] such that a(p) = 2p(k-1)/2 cos(θp).Let π(x) = ≠ {p ≤ x}. For a given subinterval [α, β] ⊂ [0, π], the now-proven Sato-Tate conjecture tells us that as x → ∞. # {p ≤ x: θp ∈ I} ~ μST([α, β]) π (x), μST([α, β]) = ∫βα 2\πsin2(θ) dθ.Let ∈ > 0. Assuming that the symmetric power L-functions of f are automorphic, we prove that as x → ∞, #{p ≤ x: θp ∈ I} = μST([α, β])π(x) + O (x/(log x)9/8- ∈) where the implied constant is effectively computable and depends only on k,N, and ∈.
Original language | English (US) |
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Pages (from-to) | 147-156 |
Number of pages | 10 |
Journal | Archiv der Mathematik |
Volume | 103 |
Issue number | 2 |
DOIs | |
State | Published - Aug 2014 |
Externally published | Yes |
Keywords
- Automorphic forms
- Sato-Tate conjecture
ASJC Scopus subject areas
- General Mathematics