### On Goldbach's problem

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We show that the Duffin and Schaeffer conjecture holds in all dimensions greater than one.

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 $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\right(x),x)]$. 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.

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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