Related papers: A sum involving the greatest-integer function

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…

Take an odd number x >0. Then 3x+1 is even and one can find an integer k> 0 so that y= 3x+1/2^k is again odd. We get in this way the mapping T, Tx=y. The paper contains two theorems describing statistical properties of T. The first…

We represent the sums $\sum_{k=0}^{n-1}{n \choose k}^{-2}$, $\sum_{k=0}^m{m\choose k}^{-1}{a\choose n-k}^{-1}$, $\sum_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\genfrac{[}{]}{0pt}{}{n}{k}}_q}$, and the sum of the reciprocals of the summands in Dixon's…

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta <…

A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…

In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2\to 2n\to 2^n\to…

Let $b > a > 0$. We prove the following asymptotic formula $$\sum_{n\ge 0} \big\lvert\{x/(n+a)\} - \{x/(n+b)\}\big\rvert = \frac{2}{\pi}\zeta(3/2)\sqrt{cx} + O(c^{2/9}x^{4/9}),$$ with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$…

Let $n,d$, and $k$ be positive integers where $n$ and $d$ are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval $[kd,\infty)$ of positive integers into a family of finite sets $X$ for which the sum…

Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to…

Let $t$ be a fixed parameter and $x$ some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{<}{>}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{<}{>}{0pt}{}{x}{k}=…

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\log\Gamma\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\…

Let $\Lambda$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\Lambda\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will…

Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of…

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

In this paper we introduce a technique to determine the sumset $A+A$, where $A$ is the indicator function of the 0's occurring in a fixed point $x$ of a substitution on the alphabet $\{0,1\}$.