A lattice sum involving the cosine function

In this article we prove that, as n→∞,∑j,k=1n−113−cos⁡2πjn−cos⁡2πkn−cos⁡2π(j+k)n∼n2log⁡n3π. We also obtain the secondary term of size ≍n2 to be followed by an error term of size O(log⁡n). In this work, results and techniques from classical analysis involving roles by several special functions, from number theory, and from numerical analysis are needed.