Sums of Three Prime Squares

Hiroshi Mikawa; Temenoujka Peneva

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 549-558
  • ISSN: 0392-4033

Abstract

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Let A , ϵ > 0 be arbitrary. Suppose that x is a sufficiently large positive number. We prove that the number of integers n ( x , x + x θ ] , satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is x θ ( log x ) - A , provided that 7 16 + ϵ θ 1 .

How to cite

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Mikawa, Hiroshi, and Peneva, Temenoujka. "Sums of Three Prime Squares." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 549-558. <http://eudml.org/doc/290367>.

@article{Mikawa2007,
abstract = {Let $A, \epsilon > 0$ be arbitrary. Suppose that $x$ is a sufficiently large positive number. We prove that the number of integers $n \in (x, x+x^\theta]$, satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is $\ll x^\theta(\log x)^\{-A\}$, provided that $\frac\{7\}\{16\} + \epsilon \leq \theta \leq 1$.},
author = {Mikawa, Hiroshi, Peneva, Temenoujka},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {549-558},
publisher = {Unione Matematica Italiana},
title = {Sums of Three Prime Squares},
url = {http://eudml.org/doc/290367},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Mikawa, Hiroshi
AU - Peneva, Temenoujka
TI - Sums of Three Prime Squares
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 549
EP - 558
AB - Let $A, \epsilon > 0$ be arbitrary. Suppose that $x$ is a sufficiently large positive number. We prove that the number of integers $n \in (x, x+x^\theta]$, satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is $\ll x^\theta(\log x)^{-A}$, provided that $\frac{7}{16} + \epsilon \leq \theta \leq 1$.
LA - eng
UR - http://eudml.org/doc/290367
ER -

References

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  7. LIU, J. Y. - ZHAN, T., On a theorem of Hua, Arch. Math. (Basel), 69 (1997), 375-390. Zbl0898.11038MR1473092DOI10.1007/s000130050136
  8. MIKAWA, H., On the sum of three squares of primes, In: Analytic Number Theory (Kyoto, 1996), 253-264, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press, Cambridge, 1997. Zbl0906.11052MR1694995DOI10.1017/CBO9780511666179.017
  9. PERELLI, A. - PINTZ, J., Hardy-Littlewood numbers in short intervals, J. Number Theory, 54 (1995), 297-308. Zbl0851.11056MR1354054DOI10.1006/jnth.1995.1120
  10. PRACHAR, K., Primzahlverteilung, Springer-Verlag, Berlin-New York, 1978. MR516660
  11. SCHWARZ, W., Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen, II, J. Reine Angew. Math., 206 (1961), 78-112. MR126431DOI10.1515/crll.1961.206.78

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