On an average ternary problem with prime powers
- PDF / 312,787 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 22 Downloads / 195 Views
On an average ternary problem with prime powers Marco Cantarini1 · Alessandro Gambini2 · Alessandro Languasco3 · Alessandro Zaccagnini4 Received: 8 October 2018 / Accepted: 3 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We continue our work on averages for ternary additive problems with powers of prime numbers in Languasco and Zaccagnini (J Number Theory 159:45–58, 2016; Rocky Mountain J Math arXiv:1806.04934, 2018) and Cantarini et al. (Proc Amer Math Soc arXiv:1805.09008, 2018). Keywords Waring–Goldbach problem · Hardy–Littlewood method Mathematics Subject Classification Primary 11P32 · Secondary 11P55 · 11P05
The author Marco Cantarini gratefully acknowledges support from a Grant “Ing. Giorgio Schirillo” from Istituto Nazionale di Alta Matematica.
B
Alessandro Zaccagnini [email protected] Marco Cantarini [email protected] Alessandro Gambini [email protected] Alessandro Languasco [email protected]
1
Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy
2
Dipartimento di Matematica Guido Castelnuovo Sapienza Università di Roma, Piazzale Aldo Moro, 5, 00185 Roma, Italy
3
Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121 Padua, Italy
4
Dipartimento di Scienze, Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze, 53/a, 43124 Parma, Italy
123
M. Cantarini et al.
1 Introduction The problem of representing a large integer n, satisfying suitable congruence conditions, as a sum of a prescribed number of powers of primes, say n = p1k1 + · · · + psks , is classical. Here k1 , …, ks denote fixed positive integers. This class of problems includes both the binary and ternary Goldbach problem, and Hua’s problem. If the density ρ = k1−1 + · · · + ks−1 is large and s ≥ 3, it is often possible to give an asymptotic formula for the number of different representations the integer n has. When the density ρ is comparatively small, the individual problem is usually intractable and it is reasonable to turn to the easier task of studying the average number of representations, if possible considering only integers n belonging to a short interval [N , N + H ], say, where H ≥ 1 is “small.” Here we study ternary problems: let k = (k1 , k2 , k3 ) where k1 , k2 and k3 are integers with 2 ≤ k1 ≤ k2 ≤ k3 . Our goal is to compute the average number of representations of a positive integer n as p1k1 + p2k2 + p3k3 , where p1 , p2 and p3 are prime numbers (or powers of primes). Let
R(n; k) =
(m 1 )(m 2 )(m 3 ),
(1)
k k k n=m 11 +m 22 +m 33
where is the von Mangoldt function, that is, ( p m ) = log( p) if p is a prime number and m is a positive integer, and (n) = 0 for all other integers. For brevity, we write ρ = k1−1 + k2−1 + k3−1 for the density of this problem. It will also shorten our formulae somewhat to write γk = (1 + 1/k) for any real k > 0, where is the Euler Gamma-function. Theorem 1.1 Let k =
Data Loading...
