Related papers: Quadratic characters with positive partial sums

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $. This result essentially is proved -- with the same tools -- by Mei-Chu Chang. See in J. Funct. Anal. 207 (2004), no 2,…

It is shown that for any prime $p$ and any natural numbers $\ell, m,$ and $s$ such that $0<s<p$, the three following congruences \begin{align*}\sum_{i\ge \ell+1}(-1)^{m-i} {m \choose i}{m+s-1+i(p-1) \choose m+s-1+\ell(p-1)} &\equiv 0 \bmod…

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in…

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and…

A few elementary estimates of a basic character sum over the prime numbers are derived here. These estimates are nontrivial for character sums modulo large q. In addition, an omega result for character sums over the primes is also included.

We study the problem of finding an universal estimation scheme $h_n:\mathbb{R}^n\to \mathbb{R}$, $n=1,2,...$ which will satisfy \lim_{t\rightarrow\infty}{\frac{1}{t}}\sum_{i=1}^t|h_ i(X_0,X_1,...,X_{i-1})-E(X_i|X_0,X_1,...,X_{i-1})|^p=0…

We study the $l^p$ norms of a class of weighted mean matrices whose diagonal terms are given by $n^{\alpha}/\sum^{n}_{i=1}i^{\alpha}$ with $\alpha > -1$. The $l^p$ norms of such matrices are known for $p \geq 2, (\alpha+1)p >1$ and $1<p…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we…

Let $(\lambda_n)_{n \geq 1}$ be a non-negative sequence with $\lambda_1>0$ and let $\Lambda_n=\sum^n_{i=1}\lambda_i$. We study the following Copson inequality for $0<p<1$, $L>p$, \begin{align*} \sum^{\infty}_{n=1}\left (\frac 1{\Lambda_n}…

When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n \le X}f(n)/n$ with error term $O((\log…

Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\alpha$ and $m\in\{1,2,3\}$, we have \begin{align*}…

A universally L^1 good sequence n_k is constructed with n_{k+1}-n_k tending to infinity. For ergodic transformations non-conventional ergodic averages of L^1 functions computed by using this sequence converge to the integral of the…

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the…

Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to…

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

We prove $q$-variation estimates, $q>2$, on $\ell^{p}$ spaces for averages along primes (with $1<p<\infty$) and polynomials (with $\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}$, where $d$ is the degree of the polynomial). This improves…

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…