Related papers: Thinned Wallis-type prime products in residue clas…

Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences…

Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$…

We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula, \begin{align*} z^{k-j}z_p^{j}=&\frac{p_{\chi}^{k-j}-1}{p_{\chi}^{k}-1}F_p(k,j;\tau)+…

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard…

By considering a limiting form of the q-Dixon_4\phi_3 summation, we prove a weighted partition theorem involving odd parts differing by >= 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of…

Let $p$ be an odd prime integer, $F/\mathbb{Q}$ be an imaginary quadratic field, and $\Psi$ be a small slope cuspidal Bianchi modular form over $F$ which is non-ordinary at $p$. In this article, we first construct a $p$-adic distribution…

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can…

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+\psi(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound…

The distribution of values of the full ranks of marked Durfee symbols is examined in prime and nonprime arithmetic progressions. The relative populations of different residues for the same modulus are determined: the primary result is that…

We revisit the proposed equality between discrete Fourier transforms of $p$-adic $\Gamma_p$--values and $p$-adic $L$--derivatives for odd characters modulo a prime $p$. The clean identity is false in general. Building on Coleman reciprocity…

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…

For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall…

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{(M|t|)^{1/3+2\delta/3},M^{2/5}|t|^{-9/20},…

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two…

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb…

This article extends our previous study on the summatory behavior of Euler's totient function $\varphi(n)$. We investigate two complementary restricted sums, $\Upsilon(x,p)=\sum_{\substack{k\le x\\\gcd(k,p)=1}}\varphi(k)$ and…