Related papers: Sur la variation de certaines suites de parties fr…

We give an asymptotic formula for the quadratic total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.

Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.

Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 3$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X +…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

The paper compares the asymptotic of the expressions $\frac {1} {x} \sum\limits_{n \leq x} {f(n)}$ and $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$, $\frac {1} {x} \sum\limits_{p \leq x} {f(p)}$ and $\sum\limits_{p \leq x} {\frac {f(p)}…

We give an asymptotic formula for the total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{\kappa_j\le T}\alpha_j H_j(1/2) = ({T\over\pi})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$…

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. Let SF(G) denotes the set of all sum-free subets of $G$ and $\sigma(G)$ denotes…

Let $R_{m, \mathrm{sq-full}}(N)$ be a representation function for the sum of a prime and a square-full number. In this article, we prove an asymptotic formula for the sum of $R_{m, \mathrm{sq-full}}(N)$ over positive integers $N$ in a short…

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^2(p)$, where $p$ runs over…

We study the sum of divisors of the quadratic form $m_1^2+m_2^2+m_3^2$. Let $$S_3(X)=\sum_{1\le m_1,m_2,m_3\le X}\tau(m_1^2+m_2^2+m_3^2).$$ We obtain the asymptotic formula $$S_3(X)=C_1X^3\log X+ C_2X^3+O(X^2\log^7 X),$$ where $C_1,C_2$ are…

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is…

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_\nu(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-\nu} K_\nu(an^2)\qquad (|\arg\,a|<\pi/2)\] as the parameter $|a|\to 0$. It…

\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+\alpha)$--order fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+\alpha}x+a(t)x=0$, $t>0$, under some simple restrictions on the…

Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is…

For "almost all" sufficiently large $N,$ satisfying necessary congruence conditions and $k\geq 2$, we show that there is an {\bf asymptotic formula} for the number of solutions of the equation \begin{align*} \begin{split}…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…