Related papers: On the expressive power of mod-$p$ linear forms on…

Let a ``complex probability'' be a normalizable complex distribution $P(x)$ defined on $\R^D$. A real and positive probability distribution $p(z)$, defined on the complex plane $\C^D$, is said to be a positive representation of $P(x)$ if…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a…

A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density $1/2$. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with…

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable $p$-adic representations of the absolute Galois groups of $p$-adic fields under the assumptions that $p$ is odd and the coefficients…

We define A_n=\sum_{i=1}^n (-1)^i\frac{1}{i} and we show that, for every prime p, there exists a number n such that A_n\equiv 0 (mod p).

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of)…

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…

Let \q be a simple algebraic group of type A or C over a field of good positive characteristic. We show for any x \in \q =\Lie(Q) that the invariant algebra S(\q_x)^{\q_x} is generated by the p^{th} power subalgebra and the mod p reduction…

Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ and $S_{i,j}$ be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over $\mathbb{Q}$ of the…

We examine canonical bases for weakly holomorphic modular forms of weight $0$ and level $p = 2, 3, 5, 7, 13$ with poles only at the cusp at $\infty$. We show that many of the Fourier coefficients for elements of these canonical bases are…

We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symmetric polynomials of even degree $2p$, the positivity of…

We introduce some (p,q)-deformations of the weight multiplicities for the representations of any simple Lie algebra g over the complex numbers. This is done by associating the indeterminate q to the positive roots of a parabolic subsystem…

For a restricted Lie superalgebra g over an algebraically closed field of characteristic p > 2, we generalize the deformation method of Premet and Skryabin to obtain results on the p-power and 2-power divisibility of dimensions of…

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$…

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by…

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there…

For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$,…