Related papers: Distinguishing newforms

Let $k$ be a positive integer such that $k\equiv3\mod4$, and let $N$ be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace $S_{\frac{k}{2}}(\Gamma_{0}(4N),F)$ of half-integral weight modular…

Let $r \ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products $\prod_{p \mid n} (p+a)$ (over primes $p$) are…

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) =…

We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_\mu$ with either $(\mu,2m_1m_2)=1$ or $(\mu,2m_1m_2)\nmid 2m_2$. As an application,…

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Then for any given $K\geq 2$, small $\epsilon>0$ and sufficiently large (only depending on $K$ and $\epsilon$) $x$, there exist at least $x^{1-\epsilon}$ integers…

Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in…

We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 +…

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n)…

Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and $F$ is a cubic form with real coefficients in $s$ variables that splits into $r$ parts, then $F$ takes arbitrarily small values at nonzero integral points. We bound…

Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we…

For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number $w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever $\{1,2,...,n\}$ is partitioned into $r$ sets $S_{0},S_{1},...,S_{r-1}$, there is some…

Let $\tau$ denote the Ramanujan tau function. One is interested in possible prime values of $\tau$ function. Since $\tau$ is multiplicative and $\tau(n)$ is odd if and only if $n$ is an odd square, we only need to consider $\tau(p^{2n})$…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

Let $F$ be a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $P(T) \in O_F[T]$ a monic separable polynomial such that $P(0) \not=0$ and $P(1) \not=0$. We give precise sufficient conditions on a given positive integer $k$…

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\bar\rho_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of primes not dividing $Np$.…

We prove new large sieve inequalities for the Fourier coefficients $\rho_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

For each positive integer n, if the sum of the factors of n is divided by n, then the result is called the abundancy index of n. If the abundancy index of some positive integer m equals the abundancy index of n but m is not equal to n, then…