arXiv subjects
Publications and source records attributed to Saurabh Kumar Singh.
Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it \right) \ll_{f, ε} \left( 1 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$
Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \mathbb{Z})$ . Let $χ$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and $r\equiv 0 (\textrm{mod} \ 3)$. Then we have \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{1/3 +ε}, \] where $ε> 0$ is any positive real number. 2. Suppose $χ$ factorizes as $χ= χ_1 χ_2$, where $ χ_i$'s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -ε} \ll P_i \ll P^{1/2 + ε}$ for $i=1,2$ and $P=P_1 P_2$. We have the Burgess bound \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{3/8 +ε}, \] where $ε> 0$ is any positive real number.
Let $k\geq 1$ be an integer. Let $δ_k(n)$ denote the maximum divisor of $n$ which is co-prime to $k$. We study the error term of the general $m$-th Riesz mean of the arithmetical function $δ_k(n)$ for any positive integer $m \ge 1$, namely the error term $E_m(x)$ where \[ \frac{1}{m!}\sum_{n \leq x}δ_k(n) \left( 1-\frac{n}{x} \right)^m = M_{m, k}(x) + E_{m, k}(x). \] We establish a non-trivial upper bound for $\left | E_{m, k} (x) \right |$, for any integer $m\geq 1$.
Let $λ_i (n)$ $i= 1, 2, 3$ denote the normalised Fourier coefficients of holomorphic eigenform or Maass cusp form. In this paper we shall consider the sum: \[ S:= \frac{1}{H}\sum_{h\leq H} V\left( \frac{h}{H}\right)\sum_{n\leq N} λ_1 (n) λ_2 (n+h) λ_3 (n+ 2h)W\left( \frac{n}{N} \right), \] \noindent where $V$ and $W$ are smooth bump functions, supported on $[1, 2]$. We shall prove a nontrivial upper bound, under the assumption that $H\geq N^{1/2+ ε}$.