Related papers: Inequalities for the overpartition function arisin…
Rosser and Schoenfeld remarked that the product $\prod_{p\leq x}(1-1/p)^{-1}$ exceeds $e^{\gamma} \log x$ for all $2\leq x\leq 10^8$, and raised the question whether the difference changes sign infinitely often. This was confirmed in a…
Almost nothing is known about the parity of the partition function $p(n)$, which is conjectured to be random. Despite this expectation, Ono surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4…
Recently, Chen, He, Hu and Xie considered the parity of the number of non-overlined (resp. overlined) parts of size greater than or equal to the size of the smallest overlined (resp. non-overlined) part in an overpartition. In this article,…
In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to…
In this paper, we provide a sublinear function $p$ on ordered Banach spaces, which depends on the order structure of the space. With respect to this $p$, we study the relation between $p$-contractivity of positive semigroups and the…
Denote by $\P_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in \P_{n_1}, \dots, D_k \in \P_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in \P_n$ according to $(n_1, \dots, n_k)$ so…
The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If…
Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…
In this paper, we introduce a subclass of p-valent non-bazilavec functions of order. Some subordination relations and the inequality properties of p-valent functions are discussed. The results presented here generalize and improve some…
In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash…
Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to…
We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell--Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the…
We present a function that tests for primality, factorizes composites and builds a closed form expression of $\pi(n^2)$ in terms of $\sum_{3 \leq p \leq n} \frac{1}{p}$ and a weaker version of $\omega(n)$.
In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
The primary focus of this paper is overpartitions, a type of partition that plays a significant role in $q$-series theory. In 2006, Treneer discovered an explicit infinite family of congruences of overpartitions modulo $5$. In our research,…
Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$…
Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders…
The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{\Gamma(\frac{n}{2}-\frac{s}{p-1})\Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1})\Gamma(\frac{n-2s}{2}-\frac{s}{p-1})}…
Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the…