Third Italian Number Theory Meeting

# Zeros of combinations of Euler products for $$\sigma>1$$

speaker: Mattia Righetti (Università di Genova)

abstract: It is well known that linear combinations of L-functions may not satisfy the Riemann Hypothesis. For example, in 1936 Davenport and Heilbronn showed that the Hurwitz zeta function $$\zeta(s,a)$$ has infinitely many zeros for $$\sigma > 1$$ when $$a$$ is transcendental or rational with $$a \neq \frac12$$ , and Cassels showed that the same is true when $$a$$ is irrational algebraic. Note that when $$a$$ is rational, $$\zeta(s,a)$$ is a linear combination of Dirichlet L-functions. Another example is given by the Epstein zeta function $$\zeta(s,Q)$$ associated to a positive definite binary quadratic form $$Q$$. Also for these functions Davenport and Heilbronn showed that they have infinitely many zeros for $$\sigma > 1$$ when the class number $$h(D)$$ of the quadratic field $$\mathbb{Q}[ \sqrt{D}]$$, where $$D$$ is the discriminant of $$Q$$, is $$> 1$$. Again, note that if $$h(D) > 1$$ then $$\zeta(s,Q)$$ may be written as a non-trivial linear combination of Hecke L-functions.

The talk will be on recent developments in this direction, namely we will show for many classes of L-functions with an Euler product that non-trivial linear and non-linear combinations have always zeros for $$\sigma > 1$$.

Moreover, we will discuss and show some results on the related problem about the distribution of the real parts of such zeros for $$\sigma > 1$$.

timetable:
Tue 22 Sep, 16:30 - 16:55, Aula Russo
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