Related papers: Lower bound for cyclic sums of Diananda type

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square…

Upper and lower bounds are derived for the quantity in the title, which is tabulated for modest values of $n$ and $k.$ An application to graphs with many cycles is given.

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(\delta\geq1)}$. We prove that if $m=\tfrac n2(\log…

We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive…

For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the…

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient…

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial…

Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x<n\leq x+H_1}d_k(n)d_l(n+h)$. More precisely, let…

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that…

Let $M_{n,r}=(\sum_{i=1}^{n}q_ix_i^r)^{\frac {1}{r}}, r \neq 0$ and $M_{n,0}=\lim_{r \rightarrow 0}M_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i$ with $q_i > 0$ satisfying $\sum^n_{i=1}q_i=1$. For a real number…

Let $f$ be a polynomial $f$ of degree $d\ge 2$ with integer coefficients which is irreducible over the rationals. Cilleruelo conjectured that the least common multiple of the values of the polynomial at the first $N$ integers satisfies…

We find an explicit upper bound for general $L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $L$-functions and Dedekind zeta functions.…

We study limit cycles of nonlinear oscillators described by the equation $\ddot x + \nu F(\dot x) + x =0$. Depending on the nonlinearity this equation may exhibit different number of limit cycles. We show that limit cycles correspond to…

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

In this paper we provide an explicit bound for $|\zeta(1+it)|$ in the form of $|\zeta(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right)$. This improves on the current best-known explicit bound of…

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll…

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg…

In this paper we classify all the cyclic finite dimensional indecomposable\\ modules of the perfect Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$ with its standard…