Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is
the Ramanujan $\tau$-function. This is equivalent to the assertion
that $\tau(n) \neq 0$ for any $n$. A related problem is to find the
distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ }
(\text{mod } p)$. These are open problems. However, the variant of
estimating the number of integers $n$ for which $n$ and $\tau(n)$
do not have a non-trivial common factor is more amenable to study.
More generally, let $f$ be a normalized eigenform for the Hecke
operators of weight $k \geq 2$ and having rational integer Fourier
coefficients $\{a(n)\}$. It is interesting to study the quantity
$(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for
Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients
$a(n)$
\begin{equation}
\{ n \leq x \text { } | \text{ } (n,a(n))=1\} =
\displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}}
\end{equation}
for some constant $U_f$. We extend this result to higher weight
forms. \\
We also show that
\begin{equation}
\{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll
\displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log
\log x}}.
\end{equation}
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/35874 |
| Date | 08 August 2013 |
| Creators | Laptyeva, Nataliya |
| Contributors | Murty, V. Kumar |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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