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In this paper, we consider the $k$-th Riesz mean for the coefficients of the Rankin-Selberg $L$-function $L_{f \times f}(s)$ related to the Godement-Jacquet $L$-function with respect to $SL(n,\mathbb{Z})$. We establish an asymptotic formula for the $k$-th Riesz mean with an improved range and a better error term. As a result, we get an asymptotic relation for the partial sum of the coefficients of $L_{f \times f}(s)$.

Let $f \in S_{k}(SL_2(\mathbb{Z}))$ be a normalized Hecke eigenforms of integral weight $k$ for the full modular group. In the article, we study the average behaviour of Fourier coefficients of $\ell$-fold product $L$-function. More precisely, we establish the asymptotics of power moments associated to the sequence $\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n- {\rm squarefree}}$ where ${f \otimes f \otimes \cdots \otimes_{\ell} f}$ denotes the $\ell$-fold product of $f$. As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences for odd $\ell$-fold product $L$-function. A similar result also holds for the sequence $\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n \in \mathbb{N}}$., Comment: 13 pages, Any comments or suggestions are welcome

In this paper, we establish a mean value result of arithmetic functions over shorter intervals with the Selberg-Delange method using the Hooley-Huxley contour.

In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier coefficients of the $j^{th}$ ($j \geq 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,sym^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over a certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$\sum_{\stackrel{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}\lambda^{2}_{sym^{j}f}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),$$ where $x$ is sufficiently large, and $$L(s,sym^{j}f):=\sum_{n=1}^{\infty}\dfrac{\lambda_{sym^{j}f}(n)}{n^{s}}.$$ When $j=2$, the error term which we obtain, improves the earlier known result.

In this paper, we study non-trivial upper bounds for the sum $\sum \limits_{n \in S} |\lambda_f(n)|$ where $f$ is a normalized Maass eigencusp form for the full modular group, $\lambda_f(n)$ is the $n$th normalized Fourier coefficient of $f$ and $S$ is a proper subset of positive integers in $[1,x]$ with certain properties.

In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement--Jacquet $L$-function $L_f(s)$ assuming the Riemann hypothesis for $L_f(s)$ and Rudnick--Sarnak conjecture.

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_{n\leq x}d^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\over 5}+\epsilon}), $$ where $\epsilon$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\over 2}+\epsilon}). $$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\over 2}(\log x)^5\log\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $$ E(x)=O(x^{1\over 2}(\log x)^5). $$, Comment: This is a revised version

For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$., Comment: 7 pages