OPEN PROBLEMS ON EXPONENTIAL AND CHARACTER SUMS

Abstract

This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of self-references in the bibliography. By no means should this be considered as an indication of anything else than the fact that the choice of problems reflects the taste and interests of the author. Thus it is not surprising that many (but not all) of the problems are related to some previous results of the author. One can find the necessary background and a wealth of information about exponential and character sums in [90] (mainly in the context of finite fields) and in [80] (in a more general contexts), see also [86, 88, 112, 131]. Some of the problems below are likely to be very difficult, some may constitute a feasible long term project (such as doctoral thesis), some are not so hard and may be used by the beginners in this area as an entry point to gain some immediate “hands-on” experience with exponential and character sums. In particular, part of the motivation for writing this list of problems has been the frequent requests to the author to suggest a research problem on exponential sums. One can also notice that most of the problems fall into two major categories. In one category we talk about obtaining new bounds of exponential