1,720,980 research outputs found
On Ramanujan expansions with additive coefficients
No full textThe Ramanujan series attached to a complex-valued arithmetic function in a fixed integer is the series , where is the so-called Ramanujan sum. Assuming that is additive or, more generally, a product of a multiplicative function with an additive one, we study the relationships between the Ramanujan series attached to in a positive integer and its subseries obtained by taking the terms with coprime to a fixed integer
On a binary diophantine inequality involving prime numbers
No full textLet 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all eta is an element of (N, 2N] \ A with \A\ = O (N exp ( -1/3 ( L/c ) 1/5) ), the inequality \p(1)(c) + p(2)(c) - eta\ < eta(1-15/14c) L8 as solutions in primes p(1), p(2) less than or equal to N-1/c
On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression
Full text linkOn the correlations, Selberg integral and symmetry of sieve functions in short intervals, III
No full textAn arithmetic function is called a sieve function of range , if it is the convolution product of the constantly function and such that , , for , and for . Here we establish a new result on the autocorrelation of by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain non-trivial asymptotic formulae that are actually unreachable by the standard approach of the distribution of in the arithmetic progressions. Moreover, we apply our asymptotic formulae to obtain new bounds for the so-called Selberg integral and symmetry integral of , which are basic tools for the study of the distribution of in short intervals
Alcune osservazioni sulla distribuzione dei numeri di Hardy-Littlewood nelle progressioni aritmetiche
No full text- …
