This is a paper about upper bounds for Dirichlet's L-function, L(s, χ), on its critical
line (s + s¯ = 1). It is to be assumed throughout that, unless otherwise stated, the
Dirichlet character, χ, is periodic modulo a prime, r, and is not the principal character
mod r. Our main theorem below shows that, if ε > 0, then
(where A is an absolute constant), for 0 < α
= (log r)/(log t) [les ] 2/753 − ε. Somewhat
weaker bounds are obtained for other cases where 0 < α [les ] 11/180 − ε.
Note that in  it was shown that, for 0 < α [les ] 2/57,
Our main theorem is a corollary of the new bounds we prove for certain exponential
sums, S, with a Dirichlet character factor:
where M2 [les ] 2M and f(x) is a real
function whose derivatives satisfy certain conditions restricting their size.