Book Review
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MathSciNet review:
1567381
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Book Information:
Author:
R. C. Vaughan
Title:
The Hardy-Littlewood method
Additional book information:
Cambridge Tracts in Mathematics, vol. 80, Cambridge University Press, Cambridge, 1981, xii + 172 pp., $34.50.
B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105. MR 97359, DOI 10.1112/S0025579300001145
H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman; Edited and prepared for publication by T. D. Browning. MR 2152164, DOI 10.1017/CBO9780511542893
H. Davenport, Cubic forms in sixteen variables, Proc. Roy. Soc. London Ser. A 272 (1963), 285–303. MR 155800, DOI 10.1098/rspa.1963.0054
H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, J. London Math. Soc. 21 (1946), 185–193. MR 20578, DOI 10.1112/jlms/s1-21.3.185
H. Davenport and D. J. Lewis, Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557–595. MR 245542, DOI 10.1098/rsta.1969.0035
Marvin J. Greenberg, Lectures on forms in many variables, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0241358
7. G. H. Hardy and J. E. Littlewood [1919], A new solution of Waring's problem, Quart. J. Math. 48, 272-293. (See also Hardy's collected papers, vol. I, Clarendon Press, Oxford, 1966, pp. 382-403.)
8. G. H. Hardy and S. Ramanujan [1918], Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2) 17, 75-115.
9. D. Hilbert [1909], Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem), Göttinger Nachrichten, 17-36.
C. Hooley, On a new approach to various problems of Waring’s type, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 127–191. MR 637346
11. L. K. Hua [1938], On Waring's problem, Quart. J. Math. 9, 199-202.
Jun-ichi Igusa, Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546292
Yu. V. Linnik, All large numbers are sums of a prime and two squares (A problem of Hardy and Littlewood). I, Mat. Sb. (N.S.) 52 (94) (1960), 661–700 (Russian). MR 0120206
Wolfgang M. Schmidt, Diophantine inequalities for forms of odd degree, Adv. in Math. 38 (1980), no. 2, 128–151. MR 597195, DOI 10.1016/0001-8708(80)90002-X
Wolfgang M. Schmidt, On cubic polynomials. I. Hua’s estimate of exponential sums, Monatsh. Math. 93 (1982), no. 1, 63–74. MR 648740, DOI 10.1007/BF01579030
16. I. M. Vinogradov [1928], Sur le théorème de Waring, C. R. Acad. Sci. USSR, 393-400.
17. I. M. Vinogradov [1937], Representation of an odd number as a sum of three primes, C. R. Acad. Sci. USSR 15, 6-7.
I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Trav. Inst. Math. Stekloff 23 (1947), 109 (Russian). MR 0029417
I. M. Vinogradov, Metod trigonometricheskikh summ v teorii chisel, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0409380
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
- 1.
- B. J. Birch [1957], Homogeneous forms of odd degree in a large number of variables, Mathematika 4, 102-105. MR 0097359
- 2.
- H. Davenport [1962], Analytic methods for diophantine equations and diophantine inequalities, Univ. of Michigan, Fall semester 1962, Campus Publishers. MR 2152164
- 3.
- H. Davenport [1963], Cubic forms in 16 variables, Proc. Roy. Soc. Ser. A 272, 285-303. MR 155800
- 4.
- H. Davenport and H. Heilbronn [1946], On indefinite quadratic forms in five variables, J. London Math. Soc. (2) 21, 185-193. MR 20578
- 5.
- H. Davenport and D. J. Lewis [1969], Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264, 557-595. MR 245542
- 6.
- M. J. Greenberg [1969], Lectures on forms in many variables, Benjamin, New York and Amsterdam. MR 241358
- 7.
- G. H. Hardy and J. E. Littlewood [1919], A new solution of Waring's problem, Quart. J. Math. 48, 272-293. (See also Hardy's collected papers, vol. I, Clarendon Press, Oxford, 1966, pp. 382-403.)
- 8.
- G. H. Hardy and S. Ramanujan [1918], Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2) 17, 75-115.
- 9.
- D. Hilbert [1909], Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem), Göttinger Nachrichten, 17-36.
- 10.
- C. Hooley [1981], On a new approach to various problems of Waring's type, Recent Progress in Analytic Number Theory (Sympos., Durham, July 1979), Academic Press, New York, pp. 127-191. MR 637346
- 11.
- L. K. Hua [1938], On Waring's problem, Quart. J. Math. 9, 199-202.
- 12.
- J. I. Igusa [1978], Lectures on forms of higher degree, Tata Inst. Fundamental Research, Bombay. MR 546292
- 13.
- Yu. V. Linnik [1960], All large numbers are sums of a prime and two squares (A problem of Hardy and Littlewood). I, Mat. Sb. (N.S.) 52 (94), 661-700. (Russian) MR 120206
- 14.
- W. M. Schmidt [1980], Diophantine inequalities for forms of odd degree, Adv. in Math. 38, 128-151. MR 597195
- 15.
- W. M. Schmidt [to appear], On cubic polynomials. II-IV; Monatsh. Math. Part I 93 (1982), 63-74. MR 648740
- 16.
- I. M. Vinogradov [1928], Sur le théorème de Waring, C. R. Acad. Sci. USSR, 393-400.
- 17.
- I. M. Vinogradov [1937], Representation of an odd number as a sum of three primes, C. R. Acad. Sci. USSR 15, 6-7.
- 18.
- I. M. Vinogradov [1947], The method of trigonometrical sums in the theory of numbers, "Nauka" Interscience, New York. MR 29417
- 19.
- I. M. Vinogradov [1971], The method of trigonometrical sums in the theory of numbers, Moscow. (Russian) MR 409380
- 20.
- H. Weyl [1916], Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77, 313-352. MR 1511862
Review Information:
Reviewer:
Wolfgang M. Schmidt
Journal:
Bull. Amer. Math. Soc.
7 (1982), 433-437
DOI:
https://doi.org/10.1090/S0273-0979-1982-15059-5
