Discrepancy of Sums of two Arithmetic Progressions

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set [N]=\{1,2,\hdots,N\} was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of $k$ ($k\geq 1$ fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form A_{1}+A_{2}+\hdots+A_{k} in $[N]$, where the $A_{i}$ are arithmetic progressions. For this hypergraph Hebbinghaus (2004) proved a lower bound of $\Omega(N^{k/(2k+2)})$. Note that the probabilistic method gives an upper bound of order $O((N\log N)^{1/2})$ for all fixed $k$. P\v{r}\'{i}v\v{e}tiv\'{y} improved the lower bound for all $k\geq 3$ to $\Omega(N^{1/2})$ in 2005. Thus, the case $k=2$ (hypergraph of sums of two arithmetic progressions) remained the only case with a large gap between the known upper and lower bound. We bridge his gap (up to a logarithmic factor) by proving a lower bound of order $\Omega(N^{1/2})$ for the discrepancy of the hypergraph of sums of two arithmetic progressions.Comment: 15 pages, 0 figure

Residue classes containing an unexpected number of primes

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly fewer primes than expected.Comment: 18 pages. Added a few remarks, changed the numbering of sections, slightly improved results, and made a few correction

Biases in prime factorizations and Liouville functions for arithmetic progressions

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of P\'olya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.Comment: 25 pages, 6 figure

The existence of small prime gaps in subsets of the integers

We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{C}$, we will establish that for any small constant $\epsilon>0$, the set $\left\{q_n| q_{n+1}-q_n \leq \epsilon \log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{C}$ has bounded prime gaps. Specific examples, such as the case where $\mathcal{C}$ is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets