## Departmental PhD Thesis Exam – Nataliya Laptyeva

Everyone is invited.  Refreshments will be served in the Math Lounge before the exam.

Thursday, August 30, 2012, 11:10 a.m., in BA 6183, 40 St. George Street

Ph.D. Candidate: Nataliya Laptyeva

Thesis Title:      A Variant of Lehmer’s Conjecture in the CM Case

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 in \cite{sanoli09} that for Hecke
eigenforms 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{L_1(x) L_3(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 \text { } | \text{ } (n,a(n)) \text{ is a prime}\} =
\displaystyle\frac{(1+o(1)) U_f x L_4^2(x)}{\sqrt{L_1(x) L_3(x)}}
\end{equation}
the number of $n \leq x$ such that $\gcd(n,a(n))$ is a prime is.