Some Related Functions to Integer GCD and Coprimality

We generalize a formula of B. Litow [Parallel Complexity of Integer Coprimality, in Electronic Colloquium on Computational Complexity, Report No. 9, 1998.] and propose several new formula linked with the parallel Integer Coprimality, Integer GCD and Modular Inverse problems as well. Particularly, we find a new trigonometrical definition of the GCD of two integers a,b⩾1:(1)gcd(a,b)=1π∫0πcos[(b−a)x]sin2(abx)sin(ax)sin(bx)dx. We also suggest a generalization of the GCD function to real numbers.