In 2012, Tseng and Yau defined the primitive cohomology groups of symplectic manifolds. The primitive cohomology includes the information from the symplectic structure, and can be used to distinguish between different symplectic structures on the same manifold. 

For any closed symplectic manifold, the alternating sum of the dimensions of its primitive cohomology groups is zero. Thus, we extract the even-degree part of its primitive cohomology and want to see whether it is relevant to any geometric information. In this talk, for any 4n-dimensional closed symplectic manifold, using the even-degree primitive cohomology groups, we define its symplectic semicharacteristic. Then, using a non-degenerate smooth vector field, we give a counting formula of the symplectic semi-characteristic. Based on this formula, we see that the symplectic semi-characteristic is an analogue of the classical Euler characteristic and is independent of the chosen symplectic form.