arXiv subjects
Publications and source records attributed to G. K. Viswanadham.
Let $\mathcal{P}$ be a subset of primes and for each prime $p\in \mathcal{P}$, consider a subset $\mathcal{L}_p$ of $\mathbb{Z}/p\mathbb{Z}$. We provide restriction estimates with integers $\leq N$ sifted by $(\mathcal{L}_p)_{\substack{p\leq z\\ p\in \mathcal{P}}}$. This generalizes a result of Green-Tao [3] on the restriction estimates.
We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston-Graham-Pintz-Yildirim \cite{GGPY}, and Maynard \cite{MAY}. An important consequence of our main theorem is existence of infinitely many pairs $α_1, α_2$ which are product of two primes in the imaginary quadratic field $K$ such that $|σ(α_1-α_2)|\leq 2$ for all embedding $σ$ of $K$ if the class number of $K$ is one and $|σ(α_1-α_2)|\leq 8$ for all embedding $σ$ of $K$ if the class number of $K$ is two.
We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $Γ_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the distribution of the signs of $\{a_f(p^m)\}_p$, where $p$ runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power $L$-functions.
In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group $SL_2(\mathbb{Z})$. We also prove a similar kind of quantitative result for the number of sign changes of the $q$-exponents $c(p) (p {vary over primes})$ of certain generalized modular functions for the congruence subgroup $Γ_0(N)$, where $N$ is square-free.