We obtain new Poisson type summation formulas with nodes $$ \pm \sqrt n and weights involving the function rk(n) that gives number of representations a positive integer as sum k squares. Our results extend due to Guinand Meyer involve sum-of-three-squares r3(n).

For $n\geqslant 5$, we prove that every $n\times n$ matrix $\mathcal{M}=(a_{i,j})$ with entries in $\{-1,1\}$ and absolute discrepancy $\lvert\mathrm{disc}(\mathcal{M})\rvert=\lvert\sum a_{i,j}\rvert\leqslant contains a zero-sum square except for the split matrices (up to symmetries). Here, is $2\times 2$ sub-matrix of $\mathcal{M}$ $a_{i,j}, a_{i+s,s}, a_{i,j+s}, a_{i+s,j+s}$ some $s\geqslant ...

The study of sums of squares in a ring or a field is a classic topic in algebra and number theory. In this context, several questions arise naturally. For example, which elements can be represented as sums of squares, and if an element can be written as a sum of squares, how many squares are actually needed ? For instance, for an integer n to be a sum of squares of integers, we must obviously h...

For every N > I we construct a set A of squares such that JA < (4/log 2)N 1 / 3 log N and every nonnegative integer n < N is a sum of four squares belonging to A . Let A be an increasing sequence of nonnegative integers and let A(x) denote the number of elements of A not exceeding x. If every nonnegative integer up to x is a sum of four elements of A, then A(x)4 > x and so A(x) > x 1/4. In 1770...

Multiplication and squaring are important operations in digital signal processing and multimedia applications. This paper presents designs for units that implement either multiplication, A × B, or sum-of-squares computations, A2 + B2, based on an input control signal. Compared to conventional parallel multipliers, these units have a modest increase in area and delay, but allow either multiplica...