23 June 2011
If two L-functions are added together, the Euler product is destroyed. Thus the linear combination is not an L-function, and hence we should not expect a Riemann Hypothesis for it. This is indeed the case: Not all the zeros of linear combinations of L-functions lie on the critical line. However, if the two L-functions have the same functional equation then almost all the zeros do lie on the critical line. This is not seen when they have different functional equations. We will discuss these results (which are due to Bombieri and Hejhal) during the talk, and demonstrate them using characteristic polynomials of random unitary matrices, where similar phenomena are observed. If the two matrices have the same determinant, almost all the zeros of linear combinations of characteristic polynomials lie on the unit circle, whereas if they have different determinants all the zeros lie off the unit circle.
- Number Theory Seminar