Short intervals asymptotic formulae for binary problems with prime powers
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 609-635.

We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms n=p 1 1 +p 2 2 and n=p 1 +m 2 , where 1 , 2 are fixed integers, p,p 1 ,p 2 are prime numbers and m is an integer.

Nous montrons des formules asymptotiques dans des intervalles courts pour le nombre moyen de représentations des entiers de la forme n=p 1 1 +p 2 2 et n=p 1 +m 2 , où 1 , 2 sont des entiers fixés, p,p 1 ,p 2 sont des nombres premiers et m est un entier.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1041
Classification: 11P32,  11P55,  11P05
Keywords: Waring-Goldbach problem, Hardy–Littlewood method
Alessandro Languasco 1; Alessandro Zaccagnini 2

1 Università di Padova Dipartimento di Matematica “Tullio Levi-Civita” Via Trieste 63 35121 Padova, Italy
2 Università di Parma Dipartimento di Scienze Matematiche Fisiche e Informatiche Parco Area delle Scienze, 53/a 43124 Parma, Italy
@article{JTNB_2018__30_2_609_0,
     author = {Alessandro Languasco and Alessandro Zaccagnini},
     title = {Short intervals asymptotic formulae for binary problems with prime powers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {609--635},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1041},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1041/}
}
TY  - JOUR
TI  - Short intervals asymptotic formulae for binary problems with prime powers
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 609
EP  - 635
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1041/
UR  - https://doi.org/10.5802/jtnb.1041
DO  - 10.5802/jtnb.1041
LA  - en
ID  - JTNB_2018__30_2_609_0
ER  - 
%0 Journal Article
%T Short intervals asymptotic formulae for binary problems with prime powers
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 609-635
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1041
%R 10.5802/jtnb.1041
%G en
%F JTNB_2018__30_2_609_0
Alessandro Languasco; Alessandro Zaccagnini. Short intervals asymptotic formulae for binary problems with prime powers. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 609-635. doi : 10.5802/jtnb.1041. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1041/

[1] Eberhard Freitag; Rolf Busam Complex analysis, Universitext, Springer, 2009, x+532 pages | Zbl: 1167.30001

[2] Alessandro Languasco; Alessandro Zaccagnini On the Hardy–Littlewood problem in short intervals, Int. J. Number Theory, Volume 4 (2008) no. 5, pp. 715-723 | Zbl: 1251.11069

[3] Alessandro Languasco; Alessandro Zaccagnini On the Hardy-Littlewood problem in short intervals, Int. J. Number Theory, Volume 4 (2008) no. 5, pp. 715-723 | Zbl: 1251.11069

[4] Alessandro Languasco; Alessandro Zaccagnini On a ternary Diophantine problem with mixed powers of primes, Acta Arith., Volume 159 (2013) no. 4, pp. 345-362 | Zbl: 1330.11063

[5] Alessandro Languasco; Alessandro Zaccagnini Short intervals asymptotic formulae for binary problems with primes and powers, II: density 1, Monatsh. Math., Volume 181 (2016) no. 2, pp. 419-435 | Zbl: 1350.11089

[6] Alessandro Languasco; Alessandro Zaccagnini Sum of one prime and two squares of primes in short intervals, J. Number Theory, Volume 159 (2016), pp. 45-58 | Zbl: 1351.11064

[7] Alessandro Languasco; Alessandro Zaccagnini Short intervals asymptotic formulae for binary problems with primes and powers, I: density 3/2, Ramanujan J., Volume 42 (2017), pp. 371-383 | Zbl: 06692048

[8] Alessandro Languasco; Alessandro Zaccagnini Sums of one prime power and two squares of primes in short intervals (2018) (https://arxiv.org/abs/1806.04934)

[9] Hugh L. Montgomery Ten lectures on the interface between analytic number theory and harmonic analysis, Regional Conference Series in Mathematics, Volume 84, American Mathematical Society, 1994, xii+220 pages | Zbl: 0814.11001

[10] Hugh L. Montgomery; Robert C. Vaughan Hilbert’s inequality, J. Lond. Math. Soc., Volume 8 (1974), pp. 73-82 | Zbl: 0281.10021

[11] Yuta Suzuki A note on the sum of a prime and a prime square, Analytic and probabilistic methods in number theory. Proceedings of the 6th international conference (Palanga, 2016), TEV, 2017, pp. 221-226

[12] Yuta Suzuki On the sum of prime number and square number (2017) (http://www.math.sci.hokudai.ac.jp/ wakate/mcyr/2017/pdf/01500_suzuki_yuta.pdf)

[13] Robert C. Vaughan The Hardy–Littlewood method, Cambridge Tracts in Mathematics, Volume 125, Cambridge University Press, 1997, vii+232 pages | Zbl: 1997

Cited by Sources: