Seminar series
Date
Thu, 23 Jun 2011
Time
16:00 - 17:00
Location
L3
Speaker
Chris Hughes
Organisation
York

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.

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