Counting in hyperbolic spikes: the diophantine analysis of   multihomogeneous diagonal equations
  A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations. 
Valentin Blomer, J\"org Br\"udern
math.NT