Exponential sums related to cusp forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T03:42:03Z http://mathoverflow.net/feeds/question/71349 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71349/exponential-sums-related-to-cusp-forms Exponential sums related to cusp forms Micah Milinovich 2011-07-26T20:06:17Z 2011-07-26T20:06:17Z <p>Let $$f(z)=\sum_{n\geq 1} a_f(n) e^{2\pi n i z}$$ be a holomorphic newform on the upper half-plane of weight $k$ for $\Gamma_0(N)$ and of trivial character which is normalized so that $a_f(1)=1$. </p> <p>In Jutila's book "A method in the theory of exponential sums" he proves an estimate of the form:</p> <p>$$\sum_{n\leq x} a_f(n) e^{2\pi i n \frac{p}{q}} \ll q^{2/3}x^{k/2-1/6+\varepsilon}$$</p> <p>where $a_f(n)$ are the coefficients of a cusp form of weight $k$ for the full modular group ($N=1)$, and $p$ and $q$ are coprime integers. Conceivably, a similar estimate holds for coefficients of holomorphic cuspforms for congruence subgroups. Does anyone know a reference for an estimate of the form:</p> <p>$$\sum_{n\leq x} a_f(n) e^{2\pi i n\frac{p}{q}} \ll_{f,q} \ \ x^{k/2-\delta}$$</p> <p>where $a_f(n)$ are the coefficients of a holomorphic cusp form of weight $k$ for $\Gamma_0(N)$, $p$ and $q$ are coprime integers, and $\delta>0$? </p>