Related papers: Rademacher's infinite partial fraction conjecture …

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix…

This diploma thesis is concerned with functional decomposition $f = g \circ h$ of polynomials. First an algorithm is described which computes decompositions in polynomial time. This algorithm was originally proposed by Zippel (1991). A…

In this paper, we calculate an exact formula for the number of partitions of a natural number $n$, where the largest part is even and no odd parts appears more than two times. The generating functions of the number of these partitions is a…

We study Rademacher processes where the coefficients are functions evaluated at fixed, but arbitrary covariables. Specifically, we assume the function class under consideration to be parametrized by the standard cocube in l dimensions and…

The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite…

Let $p_{r,s}(n)$ denote the number of partitions of a positive integer $n$ into parts containing no multiples of $r$ or $s$, where $r>1$ and $s>1$ are square-free, relatively prime integers. We use classical methods to derive a…

The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…

R. Redheffer described an $n\times n$ matrix of 0's and 1's the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and evaluate its permanent in terms of integer…

We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…

We prove a number of limiting distributions for statistics for unimodal sequences of positive integers by adapting a probabilistic framework for integer partitions introduced by Fristedt. The difficulty in applying the direct analogue of…

Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ...,…

In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived…

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not…

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers…

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$.…

We give an alternative method to that of Hardy-Ramanujan-Rademacher to derive the leading exponential term in the asymptotic approximation to the partition function p(n,a), defined as the number of decompositions of a positive integer 'n'…

We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We…

In this paper we obtain a sharp upper bound for the number of solutions to a certain diophantine inequality involving fractions with power denominator. This problem is motivated by a conjecture of Zhao concerning the spacing of such…

We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real…

We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions,…