On Goldbach's problem
Homogeneous additive equations and Waring's problem
An elementary method in prime number theory
The exceptional set of Goldbach's problem
Further improvements in Waring's problem, IV: Higher powers
Graded Lie algebras of maximal class IV
Identities in prime number theory
The -dimensional Duffin and Schaeffer conjecture
We show that the Duffin and Schaeffer conjecture holds in all dimensions greater than one.
A special case of Vinogradov's mean value theorem
On the fractional parts of and related sequences. II
As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of where is an arithmetical function (namely , , ) and is an integer (or a prime order) running over the interval . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.
On the fractional parts of and related sequences. I
This paper and its sequels deal with a new concept of distributions modulo one which is connected with the Dirichlet divisor and similar problems. Each of the theorems has some independent interest, and in addition some of the techniques developed lead to improvements in certain applications of the hyperbola method.
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