Consider the linear congruence equation $x_1+...+x_k\equiv b(mod{n}^s)$ for $b\in \mathbb{Z}$, $n,s\in \mathbb{N}$. Let ${(a,b)}_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in \mathbb{N}$ dividing $a$ and $b$ simultaneously. Let $d_1,...,d_{\tau \left(n\right)}$ be all positive divisors of $n$. For each $d_j\mid n$, define ${𝒞}_{j,s}\left(n\right)=\{1\le x\le n^s:{(x,n^s)}_s=d_j^s\}$. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ and prove that for the above linear congruence, the number of solutions...

Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu$. The $\delta$-ball $B_{\delta }$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $\mu \left(B_{2{\delta }^{\prime }}\right)\le 2^d\mu \left(B_{{\delta }^{\prime }}\right)$ for all ${\delta }^{\prime }\subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu \left(nA\right)\le n^d\mu \left(A\right)$ for all $n\ge dlogd$, then $A$ is contained in an $O\left(d{log}^{3}d\right)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.

A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.

We generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results on the behavior of a certain Witten's zeta function...