Related papers: On the Second Moment Estimate Involving the $\lamb…

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of…

In this note, we provide some results concerning the structure of a set $A\subseteq \mathbb{Z}_n^{\times}$, which has non-empty subset sums equally distributed modulo $n$. Here, $\mathbb{Z}_n^{\times}$ denotes the set which contains all the…

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…

In this paper we investigate Donkin's $(p,r)$-Filtration Conjecture, and present two proofs of the "if" direction of the statement when $p\geq 2h-2$. One proof involves the investigation of when the tensor product between the Steinberg…

We establish unconditional $\Omega$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an…

Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t…

We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related…

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a…

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a…

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With…

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $\alpha\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this…

Recent investigations on the set of commutators between the elements of a finite group having relatively prime orders have prompt us to propose a variant of the Ore conjecture: For every finite non-abelian simple group and for every $g\in…

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

We compute moments of $L$-functions associated to the polynomial family of Artin--Schreier covers over $\mathbb{F}_q$, where $q$ is a power of a prime $p>2$, when the size of the finite field is fixed and the genus of the family goes to…

A positive integer n is called a covering number if there are some distinct divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A covering…

We prove a recent conjecture due to Deutsch, Sagan, and Wilson stating that the finite sequence obtained from the first p central trinomial coefficients modulo p by replacing nonzero terms by 1's is palindromic, for any prime number p > 3.…

Let $M \subseteq \mathbb{N}_{0}$ be the additive submonoid generated by $2$ and $3$. In a recent work, Christensen, Gipson and Kulosman proved that $M$ is not a Matsuda monoid of type $2$ and type $3$ and they have raised the question of…

The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of…