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- W2952871536 abstract "Let $${f(z) = sum_{n=1}^infty a(n)e^{2pi i nz} in S_k^{mathrm{new}}(Gamma_0(N))}$$ be a newform of even weight $${k geq 2}$$ that does not have complex multiplication. Then $${a(n) in mathbb{R}}$$ for all n; so for any prime p, there exists $${theta_p in [0, pi]}$$ such that $${a(p) = 2p^{(k-1)/2} {rm cos} (theta_p)}$$ . Let $${pi(x) = #{p leq x}}$$ . For a given subinterval $${[alpha, beta]subset[0, pi]}$$ , the now-proven Sato–Tate conjecture tells us that as $${x to infty}$$ , $$ #{p leq x: theta_p in I} sim mu_{ST} ([alpha, beta])pi(x),quad mu_{ST} ([alpha, beta]) = intlimits_{alpha}^beta frac{2}{pi}{rm sin}^2(theta) dtheta. $$ Let $${epsilon > 0}$$ . Assuming that the symmetric power L-functions of f are automorphic, we prove that as $${x to infty}$$ , $$ #{p leq x: theta_p in I} = mu_{ST} ([alpha, beta])pi(x) + Oleft(frac{x}{(log x)^{9/8-epsilon}} right), $$ where the implied constant is effectively computable and depends only on k,N, and $${epsilon}$$ ." @default.
- W2952871536 created "2019-06-27" @default.
- W2952871536 creator A5081048841 @default.
- W2952871536 date "2014-08-01" @default.
- W2952871536 modified "2023-10-07" @default.
- W2952871536 title "The error term in the Sato–Tate conjecture" @default.
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- W2952871536 doi "https://doi.org/10.1007/s00013-014-0673-x" @default.
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