Jörg Brüdern
Jörg
Brüdern, Université de Stuttgart
On Weyl sums
We study the classical and simplest Weyl sum
f (α) = Σn≤N
e(αnk)
when α = a/q + β
with β and q suitably small. In this situation, a
good approximation to f is given by the product
q-1 Σ1≤c≤q
e(ack/q) ∫ [0, N]
e(βtk) dt,
and it is known that the resulting error is bounded by
O(qε
(q+qNk|β|)1/2.
It has been suggested
by various writers that this error might actually be rather
smaller, perhaps even with the exponent
1/2 in the error estimate
replaced by 1/k.
We show that this is not the case, the known bound is
essentially sharp both pointwise and in mean square.
In some cases there is an asymptotic formula
for the mean square integral of this error term.