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Let π be a self-dual irreducible cuspidal automorphic representation of GL2( A ℚ ) with trivial central character. Its Hecke eigenvalue ⁁π (n) is a real multiplicative function in n. We show that λπ (n) < 0 for some n < < Q π 2 / 5 , where Qπ denotes (a special value of) the analytic conductor. The value 2 5 is the first explicit exponent for Hecke-Maass newforms. [ABSTRACT FROM AUTHOR]

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$ th Hecke eigenvalue of  $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author ['Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms', Bull. Aust. Math. Soc.92 (2015), 195–204]. [ABSTRACT FROM AUTHOR]

We discuss in this work the distributions of values of L(1, f), where f is a primitive cusp form whose level is a prime power. We prove the upper bound part of the Montgomery-Vaughan's first conjecture and give a weaker version of the lower bound part for automorphic L-functions in this case. We establish an unweighted trace formula in aspect of prime power level in our proof. [ABSTRACT FROM AUTHOR]

Let be a self-contragredient irreducible unitary cuspidal representation of with , and be the automorphic -function attached to . Assume that is self-contragredient. Under the Generalized Ramanujan Conjecture and Generalized Riemann Hypothesis for , the estimate holds for all real number as . [ABSTRACT FROM AUTHOR]

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$ where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$. [ABSTRACT FROM AUTHOR]