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Sums of smooth squares

  • V. Blomer (a1), J. Brüdern (a2) and R. Dietmann (a3)

Abstract

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.

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References

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[1]Balog, A., On additive representation of integers, Acta Math. Hungar. 54 (1989), 297301.
[2]Blomer, V., Uniform bounds for Fourier coefficients of theta-series with arithmetic applications, Acta Arith. 114 (2004), 121.
[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), 117.
[4]Brüdern, J. and Fouvry, E., Lagrange’s four squares theorem with almost prime variables, J. Reine Angew. Math. 454 (1994), 5996.
[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), 157176.
[6]Estermann, T., On the sign of the Gaussian sum, J. Lond. Math. Soc. 20 (1945), 6667.
[7]Fouvry, E. and Tenenbaum, G., Entiers sans grand facteur premier en progressions arithmétiques, Proc. Lond. Math. Soc. (3) 63 (1991), 449494.
[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), 249273.
[9]Hall, R. R. and Tenenbaum, G., Divisors, Cambridge Tracts in Mathematics, vol. 90 (Cambridge University Press, Cambridge, 1988).
[10]Harcos, G., On sums of four smooth squares, J. Number Theory 77 (1999), 145154.
[11]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, fifth edition (Oxford University Press, New York, 1979).
[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), 159224.
[13]Hooley, C., On the representation of a number as the sum of two squares and a prime, Acta Math. 97 (1957), 189210.
[14]Hooley, C., On the Barban–Davenport–Halberstam theorem. III, J. Lond. Math. Soc. (2) 10 (1975), 249256.
[15]Hooley, C., On a new technique and its applications to the theory of numbers, Proc. Lond. Math. Soc. (3) 38 (1979), 115151.
[16]Hooley, C., On the Barban–Davenport–Halberstam theorem. X, Hardy-Ramanujan J. 21 (1998), 9 (electronic).
[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), 8695, 163 (In Russian).
[18]Neumann, S., Mean square value theorems for integers without large prime factors, PhD thesis, Universität Stuttgart (2006).
[19]O’Meara, O. T., Introduction to quadratic forms (Springer, Berlin, 1973).
[20]Plaksin, V. A., Asymptotic formula for the number of solutions of an equation with primes, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 321397, 463 (In Russian).
[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), 12451265 (In Russian).
[22]Shields, P., Some applications of sieve methods in number theory, PhD thesis, University College, Cardiff (1979).
[23]Shiu, P., A Brun–Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161170.
[24]Siegel, C. L., Über die analytische Theorie quadratischer Formen I, Ann. of Math. (2) 36 (1935), 527606.
[25]Smith, R. A., The circle problem in an arithmetic progression, Canad. Math. Bull. 11 (1968), 175184.
[26]Tenenbaum, G., Sur la probabilité qu’un entier possède un diviseur dans un intervalle donné, Compositio Math. 51 (1984), 243263.
[27]Tenenbaum, G., Introduction to analytic and probabilistic number theory (Cambridge University Press, Cambridge, 1995).
[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.
[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), 781791, 793–809.
[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), 301340.
[31]T. D., Wooley, Slim exceptional sets for sums of four squares, Proc. Lond. Math. Soc. (3) 85 (2002), 121.
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Sums of smooth squares

  • V. Blomer (a1), J. Brüdern (a2) and R. Dietmann (a3)

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