We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree $n$ and weight $k\ge n/2$ has meromorphic continuation to $ℂ$. Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight $k\ge n/2$ may be expressed in terms of the residue at $s=k$ of the associated Dirichlet series.

Let F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.

Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_1$, $k_2$ and $k_3$ for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$, respectively, and let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions ${\lambda }_g\left(n\right)$, ${\lambda }_h\left(n\right)$ twisted by ${\lambda }_f\left(n\right)$ and establish the following results: $$\sum _{n\le x}{\lambda }_f\left(n\right){\lambda }_g{\left(n\right)}^i{\lambda }_h{\left(n\right)}^j{\ll }_{f,g,h,\epsilon }{x}^{1-1/2^{i+j}+\epsilon }$$ for any $\epsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.

For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}\&lt;\sigma \&lt;1$ we consider$${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$where $R(k,\sigma ;T)=0\left(T\right)(T\to \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,\sigma ;T)$ are derived in the range min $({\beta }_k,{\sigma }_k^{*})\&lt;\sigma \&lt;1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

We have ${\sum }_{K-G\le k_j\le K+G}{\alpha }_j{H}_j^3\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ for $K^{ϵ}\le G\le K,\text{where}{\alpha }_j={\left|{\rho }_j\left(1\right)\right|}^2{(cosh\pi k_j)}^{-1},\text{and}{\rho }_j\left(1\right)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue ${\lambda }_j=k_j^2+\frac{1}{4}$ to which the Hecke series $H_j\left(s\right)$ is attached. This result yields the new bound $H_j(\frac{1}{2}{\ll }_{ϵ}{k}_j^{\frac{1}{3}+ϵ}.$