The studies of the zeta and $L$-functions of number theory possess a rich history within the subject. A natural problem that has attracted attention, and is also important for applications, is that of obtaining good upper bounds for the order of magnitude of these functions.
For rather general $L$-functions one has a functional equation relating the values at the points $s$ and $1-s$. This, together with the convexity principle of Phragmén-Lindelöf, allows one, without losing anything essential in generality, to restrict attention to the central "critical line" ${\rm Re}\,s=1/2$. It also allows one toe deduce a fairly weak but non-trivial "convexity bound". A very much stronger bound, known as the Lindelöf Hypothesis, is expected to hold, and indeed is known to follow as a simple consequence of the Generalized Riemann Hypothesis. This is the main goal of the study but, for a number of applications, it is not really necessary to have such a strong bound but is crucial to have one which improves the convexity estimate.
In the case of the Riemann zeta function, ideas developed by Voronoi, Hardy, Littlewood, Weyl, van der Corput and others early in this century led to improvements in the convexity bound. These may be generalized to a great extent to other such functions in studying the "$s$-aspect" but not to the (usually more important) problem of studying the other parameters in which these more general functions vary.
In the case of the Dirichlet $L$-functions, some fifty years later (and some thirty years ago), Burgess succeeded in breaking the convexity barrier. Although extremely important, this method too is rather special. It depends crucially on the fact that if you translate, by a small integer, the set of integers in an interval, you get something not very different from what you had at the start.
In this talk we discuss some of the history and applications of these results and go on to describe a new method which we have been developing jointly with W. Duke and H. Iwaniec which breaks the convexity bound in the above cases but also applies much more generally, for example, to the ${\rm SL}(2)$ automorphic $L$-functions, and to the $L$-functions attached to the class group of imaginary quadratic number fields.
We also discuss the related problem of finding useful approximations to $L$-functions by functions of simpler type, and especially by Dirichlet polynomials. Some basic examples of such approximations have been known for many years and furnish an important tool in the problem of obtaining bounds of the type referred to above. It would be significant, for this purpose, to find very short Dirichlet polynomials which approximate closely to the $L$-function. We discuss recent joint work with E. Bombieri which sets limits to the length of such approximations. These limits are very close to the lengths attained by approximations already known.