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Sums of smooth squares

Part of: Forms and linear algebraic groups Multiplicative number theory Additive number theory; partitions

Published online by Cambridge University Press:  03 December 2009

V. Blomer ,
J. Brüdern  and
R. Dietmann
V. Blomer
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4, USA (email: vblomer@math.toronto.edu)
J. Brüdern
Affiliation:
Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70511 Stuttgart, Germany (email: bruedern@mathematik.uni-stuttgart.de)
R. Dietmann
Affiliation:
Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70511 Stuttgart, Germany (email: dietmarr@mathematik.uni-stuttgart.de)
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Abstract

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Let R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound R(n,θ) from below.

Keywords

sums of squaressmooth numbersenveloping sieve

MSC classification

Secondary: 11P05: Waring's problem and variants 11N36: Applications of sieve methods 11E25: Sums of squares and representations by other particular quadratic forms
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1401 - 1441
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Balog, A., On additive representation of integers, Acta Math. Hungar. 54 (1989), 297–301.CrossRefGoogle Scholar
[2]Blomer, V., Uniform bounds for Fourier coefficients of theta-series with arithmetic applications, Acta Arith. 114 (2004), 1–21.CrossRefGoogle Scholar
[3]Blomer, V., Ternary quadratic forms, and sums of three squares with restricted variables, in The anatomy of integers, CRM Proceedings and Lecture Notes, vol. 46 eds A. Granville, F. Luca and J.-M. de Koninck (American Mathematical Society, Providence, RI, 2008), 1–17.Google Scholar
[4]Brüdern, J. and Fouvry, E., Lagrange’s four squares theorem with almost prime variables, J. Reine Angew. Math. 454 (1994), 59–96.Google Scholar
[5]Chen, J. R., On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176.Google Scholar
[6]Estermann, T., On the sign of the Gaussian sum, J. Lond. Math. Soc. 20 (1945), 66–67.Google Scholar
[7]Fouvry, E. and Tenenbaum, G., Entiers sans grand facteur premier en progressions arithmétiques, Proc. Lond. Math. Soc. (3) 63 (1991), 449–494.CrossRefGoogle Scholar
[8]Friedlander, J. and Lagarias, J. C., On the distribution in short intervals of integers having no large prime factor, J. Number Theory 25 (1987), 249–273.Google Scholar
[9]Hall, R. R. and Tenenbaum, G., Divisors, Cambridge Tracts in Mathematics, vol. 90 (Cambridge University Press, Cambridge, 1988).Google Scholar
[10]Harcos, G., On sums of four smooth squares, J. Number Theory 77 (1999), 145–154.Google Scholar
[11]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, fifth edition (Oxford University Press, New York, 1979).Google Scholar
[12]Heath-Brown, D. R. and Tolev, D. I., Lagrange’s four squares theorem with one prime and three almost-prime variables, J. Reine Angew. Math. 558 (2003), 159–224.Google Scholar
[13]Hooley, C., On the representation of a number as the sum of two squares and a prime, Acta Math. 97 (1957), 189–210.Google Scholar
[14]Hooley, C., On the Barban–Davenport–Halberstam theorem. III, J. Lond. Math. Soc. (2) 10 (1975), 249–256.Google Scholar
[15]Hooley, C., On a new technique and its applications to the theory of numbers, Proc. Lond. Math. Soc. (3) 38 (1979), 115–151.CrossRefGoogle Scholar
[16]Hooley, C., On the Barban–Davenport–Halberstam theorem. X, Hardy-Ramanujan J. 21 (1998), 9 (electronic).Google Scholar
[17]Kovalchik, F. B., Analogues of the Hardy–Littlewood equation, in Integral lattices and finite linear groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 86–95, 163 (In Russian).Google Scholar
[18]Neumann, S., Mean square value theorems for integers without large prime factors, PhD thesis, Universität Stuttgart (2006).Google Scholar
[19]O’Meara, O. T., Introduction to quadratic forms (Springer, Berlin, 1973).Google Scholar
[20]Plaksin, V. A., Asymptotic formula for the number of solutions of an equation with primes, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 321–397, 463 (In Russian).Google Scholar
[21]Plaksin, V. A., Asymptotic formula for the number of representations of a natural number by a pair of quadratic forms, the arguments of one of which are primes, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 1245–1265 (In Russian).Google Scholar
[22]Shields, P., Some applications of sieve methods in number theory, PhD thesis, University College, Cardiff (1979).Google Scholar
[23]Shiu, P., A Brun–Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
[24]Siegel, C. L., Über die analytische Theorie quadratischer Formen I, Ann. of Math. (2) 36 (1935), 527–606.CrossRefGoogle Scholar
[25]Smith, R. A., The circle problem in an arithmetic progression, Canad. Math. Bull. 11 (1968), 175–184.Google Scholar
[26]Tenenbaum, G., Sur la probabilité qu’un entier possède un diviseur dans un intervalle donné, Compositio Math. 51 (1984), 243–263.Google Scholar
[27]Tenenbaum, G., Introduction to analytic and probabilistic number theory (Cambridge University Press, Cambridge, 1995).Google Scholar
[28]Tolev, D., Lagrange’s four squares theorem with variables of special type, in Proc. of the session in analytic number theory and diophantine equations, Bonner Mathematische Schriften, vol. 360, eds D. R. Heath-Brown and B. Z. Moroz (Universität Bonn, 2003), 17 pp.Google Scholar
[29]Vaughan, R. C., On a variance associated with the distribution of general sequences in arithmetic progressions. I, II, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 356 (1998), 781–791, 793–809.Google Scholar
[30]Vaughan, R. C. and Wooley, T. D., Waring’s problem: a survey, in Number theory for the millennium, III, Urbana, IL, 21–26 May 2000, eds M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Philipp (A K Peters, Natick, MA, 2002), 301–340.Google Scholar
[31]T. D., Wooley, Slim exceptional sets for sums of four squares, Proc. Lond. Math. Soc. (3) 85 (2002), 1–21.Google Scholar