Related papers: Fourier Coefficients of Level 1 Hecke Eigenforms

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $\tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We…

A natural variant of Lehmer's conjecture that the Ramanujan $\tau$-function never vanishes asks whether, for any given integer $\alpha$, there exist any $n \in \mathbb{Z}^+$ such that $\tau(n) = \alpha$. A series of recent papers excludes…

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general…

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $\alpha$ is a value of $\tau(n)$. For odd $\alpha$, Murty, Murty, and Shorey proved that $\tau(n)\neq \alpha$…

Lehmer conjectured that Ramanujan's tau function never vanishes. As a variation of this conjecture, it is proved that \begin{equation*} \tau(n)\neq \pm \ell, \pm 2\ell, \pm 2\ell^2, \end{equation*} where $\ell<100$ is an odd prime, by…

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical…

In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$ never vanishes for all positive integers $m$, where the $\tau (m)$ are the Fourier coefficients of the cusp form $\Delta_{24}$ of weight 12. Lehmer verified the…

Let $\tau$ denote the Ramanujan tau function. One is interested in possible prime values of $\tau$ function. Since $\tau$ is multiplicative and $\tau(n)$ is odd if and only if $n$ is an odd square, we only need to consider $\tau(p^{2n})$…

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized…

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 100$ and all integers $m \geq 1$, we have $$ \tau(n) \neq \pm…

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author…

Let \tau(.) be the Ramanujan \tau-function, and let k be a positive integer such that \tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

Let $k$ and $n$ be positive even integers. For a Hecke eigenform $h$ in the Kohnen plus subspace of weight $k-n/2+1/2$ for $\varGamma_0(4)$, let $I_n(h)$ be the Duke-Imamoglu-Ikeda lift of $h$ to the space of cusp forms of weight $k$ for…

In this paper we significantly extend Serre's table on the vanishing properties of Fourier coefficients of odd powers of the Dedekind eta function. We address several conjectures of Cohen and Str\"omberg and give a partial answer to a…

In this paper we prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) not equal to zero for n >= 1 by investigating the additive group structure attached to tau(n) with the aid of the pigeonhole principle and unique…

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when…

An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite number $n$ such that $\varphi(n)\mid n-1$, where $\varphi$ is the Euler's totient function. It is known that if any…

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of…

We make an analytical proof for Lehmer's totient conjecture. Lehmer conjectured that there is no solution for the congruence equation $n-1\equiv 0~(mod~\phi(n))$ with composite integers,$n$, where $\phi(n)$ denotes Euler's totient function.…