Abstract
Let π be an automorphic irreducible cuspidal representation of
GL
m
{\operatorname{GL}_{m}}
over
ℚ
{\mathbb{Q}}
with unitary central character, and let
λ
π
(
n
)
{\lambda_{\pi}(n)}
be its n-th Dirichlet series coefficient.
We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded
conductor, twisted by multiplicative functions
λ
π
(
n
)
{\lambda_{\pi}(n)}
and
μ
(
n
)
λ
π
(
n
)
{\mu(n)\lambda_{\pi}(n)}
. We are able to establish non-trivial bounds for these
algebraic twisted sums with intervals of length of at
least
q
1
/
2
+
ε
{q^{1/2+\varepsilon}}
for an arbitrary fixed
ε
>
0
{\varepsilon>0}
.
On the shift semigroup on the Hardy space of Dirichlet series
