Let $f$ and $g$ be two distinct normalized primitive holomorphic cusp forms of even integral weights $k_{1}$ and $k_{2}$ for the full modular group $SL(2,\mathbb{Z})$, respectively. And let $\lambda_{f\times f\times f}(n)$ and $\lambda_{f\times f\times g}(n)$ denote the $n$th coefficients of triple product $L$-functions $L(f\times f\times f,s)$ and $L(f\times f\times g,s)$ associated to $f$ and $f,g$, respectively. In this paper, we establish the asymptotic formulae for\[\sum_{\substack{n\leq x \\ n\equiv l(\bmod q)}}\lambda_{f\times f\times f}^{2}(n) \qquad \textrm{and} \qquad \sum_{\substack{n\leq x \\ n\equiv l(\bmod q)}}\lambda_{f\times f\times g}^{2}(n)\]where $q$ is a prime with $(l,q)=1$. And we also consider other similar results of the above types.

Let $K_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$ and $h(K_n)$ the class number of $K_n$. In this paper, we claim that if an odd prime number $\ell$ satisfies $\ell\equiv3\pmod{8}$ or $\ell\equiv 5\pmod{8}$, then $\ell$ does not divide $h(K_n)$ for any positive integer $n$.

Let $X$ be a square integrable random variable with basic probability space $(\Omega, \mathcal{A}, \mathbb{P})$, taking values in the lattice $\mathcal{L}(v_0,1)=\big\{v_k=v_0+ k,k\in \mathbb{Z}\big\}$ and such that $\t_X=\sum_{k\in \mathbb{Z}}f(k)\wedge f(k+1)>0$, where $f(k)=\mathbb{P}\{X=v_k\}$, for each $k$. Let $X_i$, $i\ge 1$ be independent, identically distributed random variables having the same law as $X$, and let $S_n=\sum_{j=1}^nX_j$, for each $n$. Let $\bar{\mu}= \{\mu_k, k\in \mathbb{Z}\}$ be a sequence of non-negative reals such that $0<\mu_k<f(k) $, if $f(k)>0$, $\mu_k=0$ if $f(k)=0$, let $\mu= \sum_{k\in \mathbb{Z}}\mu_k$, and let us assume that $1- \mu < \t_X$. Let also $\bar{\tau}= \{\tau_k, k\in \mathbb{Z}\}$ be a sequence of non-negative reals satisfying the (solvable) equation: $ {\tau_{k-1}+\tau_k \over 2} = f(k) - \mu_k,$ for all $k \in \mathbb{Z}$. Let $\vartheta = \sum_{k\in \mathbb{Z}} \tau_k = 1- \mu.$ Further let $s(t) =\sum_{k\in \mathbb{Z}} \mu_k\, e^{ 2i \pi v_kt}$ and $\rho$ be such that $1-\vartheta<\rho<1$. Let $\widetilde{X}$ be the random variable associated to $X$ and $\bar{\mu}$, and defined by the relation $\mathbb{P}\{\widetilde{X} =v_k\}= {\tau_k}/{\vartheta}$, for all $k\in \mathbb{Z}$. We prove the following uniform semi-local theorems for the class $\mathcal{F}=\{F_{d}, d\ge 2\}$, where $F_{d}= d\mathbb{N}$. \begin{enumerate}[\quad (i)] \item There exists $\theta=\theta(\rho,\vartheta)$ with $ 0 < \theta < \vartheta$, $C$ and $N$ such that for $n \ge N$,\begin{multline*}\sup_{u\ge 0}\sup_{d\ge 2} \Big| \mathbb{P}\{ S_n+u\in F_{d}\} - {1 \over d}-{1 \over d}\sum_{0 <|\ell|< d }\Big(e^{(i\pi {\ell \over d}-{\pi^2\ell^2 \over 2 d^2}) } \vartheta \mathbb{E} e^{2i \pi {\ell \over d }\widetilde{X}}+s\big({\ell \over d}\big)\Big)^n \Big|\\ \le \frac{C}{\theta^{3/2}}\frac{(\log n)^{5/2}}{n^{3/2}}+2\rho^n.\end{multline*}\item Let $\mathcal{D}$ be a test set of divisors $\ge 2$, $\mathcal{D}_\varphi$ be the section of $\mathcal{D}$ at height $\varphi$ and $|\mathcal{D}_\varphi|$ denote its cardinality. Then,\[\sum_{n=2}^\infty \sup_{u\ge 0} \sup_{\varphi\ge 2} {1 \over |\mathcal{D}_\varphi|} \sum_{d\in \mathcal{D}_\varphi} \Big| \mathbb{P} \{d|S_n+u\} - {1 \over d}\Big| \le \frac{C_1}{\vartheta} + \frac{C_2}{\theta^{3/2}} + \frac{2\rho^2}{1-\rho},\]where $C_1=\frac{2e^{\pi^2/4}}{(1-e^{-\pi^2/16})}$, $C_2= C\sum_{n=2}^\infty \frac{(\log n)^{5/2}}{n^{3/2}}$.\end{enumerate}

In this paper we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface \begin{align*} x_1y_1^2+\ldots +x_sy_s^2 = 0\end{align*}in $\mathbb{P}^{s-1} \times \mathbb{P}^{s-1} $ with $s \geq 7$. This confirms the Manin conjecture for this variety.

We obtain $q$-continued fractions $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ of order fourteen, and continued fractions $V_{1}(q)$, $V_{2}(q)$ and $V_{3}(q)$ of order twenty-eight as special cases of a general continued fraction identity of Ramanujan. We prove some theta-function identities for the continued fractions and show that colour partition identities can be deduced from the theta-function identities. Some vanishing coefficients results arising from the continued fractions are also offered.

Let $\lambda$ denote the Liouville function, and let $(n,q)$ be the greatest common divisor of positive integers $n$ and $q$. For any positive real numbers $x$ and $y$, in this paper, we shall derive asymptotic formulas for the sums $\sum_{n\leq y}\l(\sum_{q\leq x}\sum_{d|(n,q)}d\lambda\l({q}/{d}\r)\r)^{k}$ in the case $k=1, 2.$

We apply the resonance method to Montgomery's convolution formula for $\operatorname{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ in the strip $1/2 < \sigma < 1$. This gives new insight into maximal values of $\operatorname{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ for $t \in [T^{\beta},T]$ for all $\beta \in (0,1)$ and real $\theta$.