Related papers: Quartic and Octic Characters Modulo n

We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(\frac{1}{2},\chi)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's…

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when…

This paper defines an asymptotic character map which is a morphism from the Grothendieck group of category $\mathcal{O}$ of an integral filtered quantization to rational functions on the Lie algebra of a torus. We show that the asymptotic…

Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method we show that L^{(k)}(1/2,chi) is not equal to zero for almost all the characters chi in C(q,+) when k and q are large. Here, L^{(k)}(s,chi) is…

In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math.…

In this paper it is proved that there are constants 0< c_2< c_1 such that an asymptotic formula can be given for the the number of (labeled) n-vertex graphs of diameter d whenever n tends to infinity and 2 < d < n - c_1 (log n). A typical…

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j,…

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand…

We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} \chi(n)$, where $\chi$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erd\H{o}s, and more recently…

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than…

We investigate a conjecture to describe the characters of large families of RCFT's in terms of contour integrals of Feigin-Fuchs type. We provide a simple algorithm to determine the modular S-matrix for arbitrary numbers of characters as a…

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.

Littlewood investigated polynomials with coefficients in $\{-1,1\}$ (Littlewood polynomials), to see how small their ratio of norms $||f||_4/||f||_2$ on the unit circle can become as $deg(f)\to\infty$. A small limit is equivalent to slow…

We describe an algorithm, which - given the characters of tilting modules and assuming that Donkin's tilting conjecture is true - computes the characters of simple modules for an algebraic group in any characteristic.

Motivated by applications to the study of L-functions, we develop an asymptotic version of the large sieve inequality for linear forms in primitive Dirichlet characters.

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this…

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which…

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the…

Let $d$ be a positive integer. We study the proportion of irreducible characters of infinite families of irreducible Coxeter groups whose values evaluated on a fixed element $g$ are divisible by $d$. For Coxeter groups of types $A_n, B_n$…