Related papers: Sums with Stern-Brocot sequences and Minkowski que…

The Minkowski question mark function, maping the unit interval to itself, is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of…

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…

In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of…

We exploit some properties of the Hurwitz zeta function $\zeta (n,x)$ in order to study sums of the form $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}1/(jk+l)^{n}$ and $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}(-1)^{j}/(jk+l)^{n}$ for $%…

In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary…

We study Birkhoff sums over rotations (series of the form $\sum_{r=1}^{N}\phi(r\alpha)$), in which the summed function $\phi$ may be unbounded at the origin. Estimates of these sums have been of significant interest and application in pure…

We give a direct combinatorial proof of a famous identity, $$ \sum_{i+j=n} m{2i}{i} \binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the…

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n,…

Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\…

For a compact set $A \subset {\mathbb R}^d$ and an integer $k\ge 1$, let us denote by $$ A[k] = \left\{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right\}=\sum_{i=1}^k A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann…

We consider the recursion $X_{n+1}=\sum_{i=0}^n \epsilon_{n,i}X_{n-i}$, where $\epsilon_{n,i}$ are i.i.d. (Bernoulli) random variables taking values in $\{-1,1\}$, and $X_0=1$, $X_{-j}=0$ for $j>0$. We prove that almost surely, $n^{-1}\log…

We prove that the category of representations of the N-Kronecker quiver and that of coherent sheaves on the noncommutative projective scheme of $R=k< X_1,...,X_N >/(\sum^N_{i=1}X_i^2)$ are derived equivalent. This equivalence is easily…

Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by $S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$ In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Z.-W. Sun…

In a landmark paper on arithmetical properties of Lambert series, Erd\H{o}s proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erd\H{o}s-Borwein constant. Crandall, in 2012, studied…

The Minkowski question-mark function $?(x)$ is a continuous strictly increasing function defined on $[0,1]$ interval. It is well known fact that the derivative of this function, if exists, can take only two values: $0$ and $+\infty$. It is…

In 2017, O. Deiser and C. Lasser obtained an explicit formula for the $n$-th derivative of the inverse tangent function. We calculate this derivative by a different method based on Fa\`a di Bruno's formula. Comparing the two results leads…

The sum of the absolute values of the Fourier coefficients of a function $f:\mathbb{F}_2^n \to \mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral…

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…

We consider the following polynomial generalization of Stern's diatomic series: let $s_1(x,y)=1$, and for $n\geq 1$ set $s_{2n}(x,y)=s_n(x,y)$ and $s_{2n+1}(x,y)=x\,s_n(x,y)+y\,s_{n+1}(x,y)$. The coefficient $[x^iy^j]s_n(x,y)$ is the number…

In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions,…