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

formula here

(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 [13] it was shown that, for 0 < α [les ] 2/57,

formula here

Our main theorem is a corollary of the new bounds we prove for certain exponential sums, S, with a Dirichlet character factor:

formula here

where M2 [les ] 2M and f(x) is a real function whose derivatives satisfy certain conditions restricting their size.