Note on Heath-Brown’s estimate for Heilbronn’s exponential sum

Mordell’s Exponential Sum Estimate Revisited

A Note on Chromatic Sum

The chromatic sum Σ(G) of a graph G is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present Σ(G) < χf (G)|G| for every graph G.

A Note on Exponential-logarithmic and Logarithmic-exponential Series

We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series (constructed in [KK1], [K] and [KS]). On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsicall...

Vinogradov’s estimate for exponential sums over primes

1Many of the manipulations of sums in these chapters are hard to follow, and I greatly expand on the calculations in Davenport. The organization of the proof in Davenport seems to be due to Vaughan. I have also used lecture notes by Andreas Strömbergsson, http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf, pp. 245–257. Another set of notes, which I have not used, are http://jonismathnote...

Note on a Zero-Sum Problem

Let G be an additively written, finite abelian group, and exp(G ) its exponent. Let S=(a1 , ..., ak) be a sequence of elements in G; we say that S is a zero-sum sequence if i=1 ai=0. Let s(G ) be the samllest integer t such that every sequence of t elements in G contains a zero-sum subsequence of length exp(G ). This constant has been studied by serveral authors during last 20 years [1 9, 11]. ...