## Application of The Circle Method in Five Number Theory Problems

##### Abstract

This thesis consists of three applications of the circle method in number theory problems. In the first chapter, we study a question of Graham.
Are there infinitely many integers $n$ for which the central binomial coefficient
$\binom{2n}{n}$ is relatively prime to $105 = 3\cdot 5 \cdot 7$?
By Kummer's Theorem,
this is the same to ask if there are infinitely many integers $n$, so that $n$ added to itself base $3$, $5$, or $7$, has no carries. A probabilistic heuristic of Pommerance predicts that there should be infinitely many such integers $n$.
We establish a result of statistical nature supporting Pommerance's heuristic.
The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes.
In the second chapter, 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 for the partition function 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 chapter of this thesis is about endpoint scale independent $\ell^p-$improving inequality for averages over the prime numbers.
The primes are almost full-dimensional, hence one expects improving estimates
for all $p >1$. Those are known, and relatively easy to establish.
The endpoint estimates are far more involved however, engaging
for instance Siegel zeros, in the unconditional case, and the
Generalized Riemann Hypothesis (GRH) in the general case. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a logarithmic factor. The proof is based on a circle method argument, and utilizing smooth numbers to gain additional control of Ramanujan sums.