19 research outputs found
Asymptotics for numbers of line segments and lines in a square grid
We present an asymptotic formula for the number of line segments connecting
q+1 points of an nxn square grid, and a sharper formula, assuming the Riemann
hypothesis. We also present asymptotic formulas for the number of lines through
at least q points and, respectively, through exactly q points of the grid. The
well-known case q=2 is so generalized
Arithmetical properties of Multiple Ramanujan sums
In the present paper, we introduce a multiple Ramanujan sum for arithmetic
functions, which gives a multivariable extension of the generalized Ramanujan
sum studied by D. R. Anderson and T. M. Apostol. We then find fundamental
arithmetic properties of the multiple Ramanujan sum and study several types of
Dirichlet series involving the multiple Ramanujan sum. As an application, we
evaluate higher-dimensional determinants of higher-dimensional matrices, the
entries of which are given by values of the multiple Ramanujan sum.Comment: 19 page