Further results on a generalization of Bertrand's postulate.
Further results on primes in small intervals.
Gaps between primes, and the pair correlation of zeros of the zeta-function
Generalizing a theorem of Schur
In a letter written to Landau in 1935, Schur stated that for any integer , there are primes such that . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers and real , there exist primes such that
Grandes valeurs des fonctions arithmétiques
Grandes valeurs d'une fonction liée au produit d'entiers consécutifs
Große Differenzen zwischen aufeinanderfolgenden Primzahlen.
Higher correlations of divisor sums related to primes. I: Triple correlations.
Index of irregularity of a prime.
Inequalities concerning the function π(x): Applications
Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while . The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for . The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe and Schoenfeld...
Infinitude of Wilson primes for
Integers without large prime factors, II
Irrégularités dans la distribution des nombres premiers dans les progressions arithmétiques
Irregularities in the distribution of primes in short intervals.
Jumping champions.
Landau’s problems on primes
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 :(1)Are there infinitely many primes of the form ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these...
Les fonctions arithmétiques et le plus grand facteur premier
Levels of Distribution and the Affine Sieve
We discuss the notion of a “Level of Distribution” in two settings. The first deals with primes in progressions, and the role this plays in Yitang Zhang’s theorem on bounded gaps between primes. The second concerns the Affine Sieve and its applications.
Local problems with primes I.
Lois de répartition des diviseurs.