Georgia Institute of Technology College of Sciences

This thesis consists of three applications of the circle method in number theory problems. In the first part, we study the $p-$divisibility of the central binomial coefficients. For a certain set of large prime numbers, we prove that there are infinitely many integers $n$, which $\binom{2n}{n}$ has these primes with unexpectedly small multiplicity in its prime factorization. This result is related to an open problem conjectured by Graham, stating that there are infinitely many integers $n$ which the binomial coefficients $\binom{2n}{n}$ is coprime with $105$. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes.

In the second part, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument.

The third part of this thesis is about finding an endpoint $\ell^p-$improving inequality for an ergodic sum involving the primes. As the set of the prime is almost full-dimensional, the question on the endpoint becomes more interesting, because we want to bound $\ell^{\infty}$ to $\ell^{1}$ operator. The weak-type inequality we propose depends on the assumption of the Generalized Riemann Hypothesis. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a $\log $ factor.  The proof is based on a circle method argument and careful use of the Ramanujan sums.