منابع مشابه
Additive Functions on Shifted Primes
Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes. A real-valued function / defined on the positive integers is additive if it satisfies f(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determined by their values on the prime-powers. For additive arithmetic function /, let Q denote the frequency...
متن کاملDivisors of Shifted Primes
Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend h...
متن کاملMultiple Recurrence and Convergence for Certain Averages along Shifted Primes
We show that any subset A ⊂ N with positive upper Banach density contains the pattern {m,m + [nα], . . . ,m + k[nα]}, for some m ∈ N and n = p − 1 for some prime p, where α ∈ R\Q. Making use for the Furstenberg Correspondence Principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averag...
متن کاملThe Polynomial Multidimensional Szemerédi Theorem along Shifted Primes
If ~q1, . . . , ~qm : Z → Z are polynomials with zero constant terms and E ⊂ Z has positive upper Banach density, then we show that the set E ∩ (E − ~q1(p− 1))∩ . . .∩(E−~qm(p−1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results ...
متن کاملCatalan Numbers, Primes and Twin Primes
with C0 = 1. Their appearances occur in a dazzling variety of combinatorial settings where they are used to enumerate all manner of geometric and algebraic objects (see Richard Stanley’s collection [28, Chap. 6]; an online Addendum is continuously updated). Quite a lot is known about the divisibility of the Catalan numbers; see [2, 10]. They are obviously closely related to the middle binomial ...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2012
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s179304211250114x