Related papers: Pure Gauss sums and skew Hadamard difference sets

By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $n\ge 1$ and $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In…

The classical Dedekind sums $s(d, c)$ can be represented as sums over the partial quotients of the continued fraction expansion of the rational $\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the transformation of the…

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived…

Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states…

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with…

We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form $1 + \frac{1}{3}+ \cdots + \frac{1}{2n-1}$. Our proof involves techniques from the theory of colored…

Let $S(z)$ be an absolutely convergent Dirichlet series with a bounded spectrum and only real zeros $a_n$, let $\mu$ be the sum of unit masses at points $a_n$. It is proven that the Fourier transform $\hat\mu$ in the sense of distributions…

We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By…

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof…

We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale…

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{\sigma(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that…

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$.…

In [3], we have introduced a probability measure to study the power and exponential sums for a certain coding system. The distribution function of the probability measure gives explicit formulas for the power and exponential sums.…

Prototypical rational vertex operator algebras are associated to affine Lie algebras at positive integer level k. They correspond physically to the Wess-Zumino-Witten theories, and their representation theory can be captured by quantum…

A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a…

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such…

A conjecture of Graham (repeated by Erd\H{o}s) asserts that for any set $A \subseteq \mathbb{F}_p \setminus \{0\}$, there is an ordering $a_1, \ldots, a_{|A|}$ of the elements of $A$ such that the partial sums $a_1, a_1+a_2, \ldots,…

The zeroes of Goss polynomials $G_{k, \Lambda}(X)$ for $\Lambda = A \defeq \mathbb{F}_{q}[T]$ and similar lattices $\Lambda$ are studied. Generically, the zero distribution follows a simple pattern governed by the $q$-adic expansion of…