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Displaying 241 – 260 of 414

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Oscillation of Mertens’ product formula

Harold G. Diamond, Janos Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

Mertens’ product formula asserts that p x 1 - 1 p log x e - γ as x . Calculation shows that the right side of the formula exceeds the left side for 2 x 10 8 . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on π ( x ) - li x , this and a complementary inequality might change their sense for sufficiently large values of x . We show this to be the case.

P 2 in short intervals

Henryk Iwaniec, M. Laborde (1981)

Annales de l'institut Fourier

For any sufficiently large real number x , the interval [ x , x + x 0 , 45 ] contains at least one integer having at most two prime factors .

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël Rivat (2015)

Journal of the European Mathematical Society

For a large class of digital functions f , we estimate the sums n x Λ ( n ) f ( n ) (and n x μ ( n ) f ( n ) , where Λ denotes the von Mangoldt function (and μ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

Primes in almost all short intervals. II

Danilo Bazzanella (2000)

Bollettino dell'Unione Matematica Italiana

In questo lavoro vengono migliorati i risultati ottenuti in «Primes in Almost All Short Intervals» riguardo la distribuzione dei primi in quasi tutti gli intervalli corti della forma g n , g n + H , con g n funzione reale appartenente ad una ampia classe di funzioni. Il problema viene trattato mettendo in relazione l'insieme eccezionale per la distribuzione dei primi in intervalli nella forma g n , g n + H con l'insieme eccezionale per la formula asintotica ψ x + H - ψ x H  as  x . I risultati presentati vengono quindi ottenuti grazie allo studio...

Currently displaying 241 – 260 of 414