"Toward Riemann's Q-hypothesis"
I will introduce q-counteparts of Riemann's zeta and the
Dirichlet L-functions. Upon the symmetrization "s <-> 1-s",
their nontrivial zeros are expected to satisfy the Riemann
hypothesis for q sufficiently close to 1; confirmations are
entirely numerical. The meromorphic continuations are based
on the technique of q-integras in terms of Macdonald's measure
function. Here Cauchy's residue theorem can be also used like
in Riemann's first proof of the functional equation. Calculating
their limits as q->1 involves the method of shift operators.
This approach is directly related to integral formulas for
canonical traces in the theory of dahas extending the approach
by Arthur, Heckman, Opdam and others to the decomposition of
the canonical trace in the affine Hecke theory (Kazhdan-Lusztig
and Lusztig).