Related papers: Dedekind Sums with Even Denominators

This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a…

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that,…

We study three restrictions on normalizers or centralizers in finite p-groups, namely: (i) |N_G(H) : H| <= p^k for every H non-normal in G, (ii) |N_G(<g>) : <g>| <= p^k for every <g> non-normal in G, and (iii) |C_G(g) : <g>| <= p^k for…

Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of…

We propose a regularization of the formal differential expression of order $m \geqslant 3$ $$ l(y) = i^my^{(m)}(t) + q(t)y(t), \,t \in (a, b), $$ applying quasi-derivatives. The distribution coefficient $q$ is supposed to have an…

We provide estimates for $s^{\rm th}$ moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author's iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all…

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a…

For the Gauss sums which are defined by S_n(a,q) := \sum_{x (mod q)} e(ax^n/q), Stechkin (1975) conjectured that the quantity A := \sup_{n,q\ge 2} \max_{\gcd(a,q)=1} |S_n(a,q)|/q^(1-1/n) is finite. Shparlinski (1991) proved that A is…

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We…

Let $A$ be a sufficiently dense subset of a finite field $\mathbb F_q$ or a finite, cyclic ring $\mathbb Z/ N\mathbb Z$. Assuming that $q$ and $N$ have no small prime divisors, we show that generalised Fermat equations have the expected…

In the process of proving a sharpened form of G\r{a}rding's inequality, Fefferman & Phong demonstrated that every non-negative function $f\in C^{3,1}(\mathbb{R}^n)$ can be written as a finite sum of squares of functions in…

The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we…

For any given integer $k\geq 2$ we prove the existence of infinitely many $q$ and characters $ \chi\pmod q$ of order $k$, such that $|L(1,\chi)|\geq (e^{\gamma}+o(1))\log\log q$. We believe this bound to be best possible. When the order $k$…

In a previous paper, I have defined non--commutative generalized Dedekind symbols for classical $PSL(2,Z)$--cusp forms using iterated period polynomials. Here I generalize this construction to forms of real weights using their iterated…

We conjecture a formula for the rational $q,t$-Catalan polynomial $\mathcal{C}_{r/s}$ that is symmetric in $q$ and $t$ by definition. The conjecture posits that $\mathcal{C}_{r/s}$ can be written in terms of symmetric monomial strings…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\chi:[1,n] \rightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$…

Distorted sums of models were introduced and discussed in [Sh:463]. This notion generalizes the notion of disjoint (or direct) sums of models by letting the summands overlap. In the first section we investigate types in distorted sums and…

Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define…