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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