**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

**Ph.D. Advisor**: V. Kumar Murty

**Thesis Title: A Variant of Lehmer’s Conjecture in the CM Case
**http://www.math.toronto.edu/graduate/ut-thesisLaptyeva.pdf

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.

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