3,114 research outputs found
The ternary Goldbach problem
The ternary Goldbach conjecture, or three-primes problem, states that every
odd number greater than can be written as the sum of three primes. The
conjecture, posed in 1742, remained unsolved until now, in spite of great
progress in the twentieth century. In 2013 -- following a line of research
pioneered and developed by Hardy, Littlewood and Vinogradov, among others --
the author proved the conjecture.
In this, as in many other additive problems, what is at issue is really the
proper usage of the limited information we possess on the distribution of prime
numbers. The problem serves as a test and whetting-stone for techniques in
analysis and number theory -- and also as an incentive to think about the
relations between existing techniques with greater clarity.
We will go over the main ideas of the proof. The basic approach is based on
the circle method, the large sieve and exponential sums. For the purposes of
this overview, we will not need to work with explicit constants; however, we
will discuss what makes certain strategies and procedures not just effective,
but efficient, in the sense of leading to good constants. Still, our focus will
be on qualitative improvements.Comment: 29 pages. To be submitted to the Proceedings of the ICM 201
On Some Dynamical Systems in Finite Fields and Residue Rings
We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning
Double Character Sums over Subgroups and Intervals
We estimate double sums with a multiplicative character
modulo where and is a subgroup of order
of the multiplicative group of the finite field of elements. A nontrivial
upper bound on can be derived from the Burgess bound if and from some standard elementary arguments if , where is arbitrary. We obtain a
nontrivial estimate in a wider range of parameters and . We also
estimate double sums and give an application to primitive
roots modulo with non-zero binary digits
Note on the Theory of Correlation Functions
The purpose of this note is to improve the current theoretical results for
the correlation functions of the Mobius sequence and the
Liouville sequence .Comment: Sixty Six Pages. Keywords: Autocorrelation function, Correlation
function, Multiplicative function, Liouville function, Mobius function, von
Mangoldt function, Exponential Su
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