×

The exceptional set in Hua’s theorem for three squares of primes. (English) Zbl 1142.11071

Summary: It is proved that with at most \(O(N^ {11/12+\varepsilon})\) exceptions, all positive integers \(n \leq N\) satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hua, L. K.: Some results in the additive prime number theory. Quart. J. Math. (Oxford), 9, 68–80 (1938) · JFM 64.0131.02 · doi:10.1093/qmath/os-9.1.68
[2] Schwarz, W.: Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen II. J. Riene Angew. Math., 206, 78–112 (1961) · Zbl 0102.28201 · doi:10.1515/crll.1961.206.78
[3] Leung, M. C., Liu, M. C.: On generalized quadratic equations in three prime variables. Mh. Math., 115, 133–169 (1993) · Zbl 0779.11045 · doi:10.1007/BF01311214
[4] Liu, J. Y., Zhan T.: Sums of five almost equal prime squares (II). Sci. China, 41, 710–722 (1998) · Zbl 0938.11048 · doi:10.1007/BF02901953
[5] Bauer, C., Liu, M. C., Zhan, T.: On sums of three prime squares. J. Number Theory, 85, 336–359 (2000) · Zbl 0961.11034 · doi:10.1006/jnth.2000.2552
[6] Liu, J. Y., Liu, M. C.: The exceptional set in the four prime squares problem. Illinois J. Math., 44, 272–293 (2000) · Zbl 0942.11044
[7] Liu, J. Y., Liu, M. C., Zhan, T.: Squares of primes and powers of two. Mh. Math., 128, 283–313 (1999) · Zbl 0940.11047 · doi:10.1007/s006050050065
[8] Ren, X. M.: The Waring–Goldbach problem for cubes. Acta Arith., XCIV, 287–301 (2000) · Zbl 0967.11041
[9] Ren, X. M.: The exceptional set in Roth’s theorem concerning a cube and three cubes of primes. Quart. J. Math. (Oxford), 52, 107–126 (2001) · Zbl 0991.11056 · doi:10.1093/qjmath/52.1.107
[10] Liu, J. Y., Liu, M. C.: Representations of even integers as squares of primes and powers of 2. J. Number Theory, 83, 202–225 (2000) · Zbl 0961.11035 · doi:10.1006/jnth.1999.2500
[11] Liu, J. Y., Liu, M. C.: Representation of even integers by cubes of primes and powers of 2. Acta Math. Hungar., 91, 217–243 (2001) · Zbl 0980.11045 · doi:10.1023/A:1010671222944
[12] Liu, J. Y., Liu, M. C., Zhan, T.: Squares of primes and powers of two (II). J. Number Theory, 92, 99–116 (2002) · Zbl 0997.11082 · doi:10.1006/jnth.2001.2689
[13] Wooley, T. D.: Slim exceptional sets for sums of four squares. Proc. London Math. Soc. (3), 85, 1–21 (2002) · Zbl 1039.11066
[14] Liu, J. Y., Zhan, T.: Distribution of integers that are sums of three squares of primes. Acta Arith., XCVIII, 207–228 (2001) · Zbl 0972.11099 · doi:10.4064/aa98-3-1
[15] Liu, J. Y.: On Lagrange’s theorem with prime variables. Quart. J. Math. (Oxford), 54, 453–462 (2003) · Zbl 1080.11071 · doi:10.1093/qmath/hag028
[16] Davenport, H.: Multiplicative Number Theory, 2nd ed., Springer, Berlin, 1980 · Zbl 0453.10002
[17] Heath–Brown, D. R.: Prime numbers in short intervals and a generalized Vaughan’s identity. Can. J. Math., 34, 1365–1377 (2003) · Zbl 0494.10027 · doi:10.4153/CJM-1982-095-9
[18] Pan, C. D., Pan, C. B.: Fundamentals of analytic number theory (in Chinese), Science Press, Beijing, 1991 · Zbl 0738.55007
[19] Titchmarsh, E. C.: The theory of the Riemann zeta–function, 2nd ed., University Press, Oxford, 1986 · Zbl 0601.10026
[20] Prachar, K.: Primzahlverteilung, Springer, Berlin, 1957
[21] Huxley, M. N.: Large values of Dirichlet polynomials (III). Acta Arith., 26, 435–444 (1974/75)
[22] Gallagher, P. X.: A large sieve density estimate near {\(\sigma\)} = 1. Invent. Math., 11, 329–339 (1970) · Zbl 0219.10048 · doi:10.1007/BF01403187
[23] Bombieri, E.: Le grand crible dans la théorie analytique des nombres, Asterisque, 18, Soc. Math. France, Paris, 1974 · Zbl 0292.10035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.