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Acta Arithmetica

### Landau’s problems on primes

Journal de Théorie des Nombres de Bordeaux

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following : Are there infinitely many primes of the form ${n}^{2}+1$?

### On the exceptional set for the $2k$-twin primes problem

Compositio Mathematica

### On the comparative theory of primes

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Über eine Verallgemeinerung des Tschebyschef-Problems.

Mathematische Zeitschrift

### Oscillation of Mertens’ product formula

Journal de Théorie des Nombres de Bordeaux

Mertens’ product formula asserts that $\prod _{p\le x}\left(1-\frac{1}{p}\right)\phantom{\rule{0.166667em}{0ex}}logx\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{e}^{-\gamma }$ as $x\to \infty$. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^{8}$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

### Primes in tuples IV: Density of small gaps between consecutive primes

We prove that given any small but fixed η > 0, a positive proportion of all gaps between consecutive primes are smaller than η times the average gap. We show some unconditional and conditional quantitative results in this vein. In the results the dependence on η is given explicitly, providing a new quantitative way, in addition to that of the first paper in this series, of measuring the effect of the knowledge on the level of distribution of primes.

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