Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T08:53:28.693Z Has data issue: false hasContentIssue false
  • English
  • Français

Hardy's legacy to number theory

Part of: Discontinuous groups and automorphic forms Additive number theory; partitions Multiplicative number theory History of mathematics and mathematicians Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  09 April 2009

R. C. Vaughan
R. C. Vaughan
Affiliation:
Address From 1st January 1999: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This is an expanded version of two lectures given at the conference held at Sydney University in December 1997 on the 50th anniversary of the death of G. H. Hardy.

MSC classification

Secondary: 01A60: 20th century 11F11: Holomorphic modular forms of integral weight 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11N35: Sieves 11P05: Waring's problem and variants 11P32: Goldbach-type theorems; other additive questions involving primes 11P55: Applications of the Hardy-Littlewood method 11P82: Analytic theory of partitions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 2 , October 1998 , pp. 238 - 266
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Baker, R. C. and Harman, G., ‘Diophantine approximation by prime numbers’, J. London Math. Soc. 25 (1982), 201–215.CrossRefGoogle Scholar
[2]Birch, B. J., ‘Waring's problem in algebraic number fields’, Proc. Cam. Phil. Soc. 57 (1961), 449–459.CrossRefGoogle Scholar
[3]Birch, B. J., ‘Forms in many variables’, Proc. Royal Soc. London Ser. A 265 (1962), 245–263.Google Scholar
[4]Boklan, K. D., ‘The asymptotic formula in Waring's problem’, Mathematika 41 (1994), 147–161.CrossRefGoogle Scholar
[5]Bombieri, E., ‘On the large sieve’, Mathematika 12 (1965), 201–225.CrossRefGoogle Scholar
[6]Bombieri, E. and Davenport, H., ‘Small differences between prime numbers’, Proc. Royal Soc. London Ser. A 293 (1966), 1–18.Google Scholar
[7]Bombieri, E. and Davenport, H., ‘On the large sieve method’, in: Abh. aus Zahlentheorie und Analysis Zur Erinnerung an Edmund Landau (eut. Verlag Wiss., Berlin, 1968) pp. 11–22.Google Scholar
[8]Bombieri, E. and Iwaniec, H., ‘On the order ζ(1/2+it)’, Ann. Scuola Norm. Pisa CI. Sci. 13 (1986), 449–472.Google Scholar
[9]Bombieri, E. and Iwaniec, H., ‘Some mean value theorems for exponential sums’, Ann. Scuola Norm. Pisa Cl. Sci. 13 (1986), 473–486.Google Scholar
[10]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’, Kexue Tongbao 17 (1966), 385–386 Foreign Lang. Ed.Google Scholar
[11]Chudakov, N. G., ‘On the Goldbach problem’, Comptes Rendus Acad. Sci. URSS 17 (1937), 335–338.Google Scholar
[12]Conrey, J. B., ‘Zeros of derivatives of Riemann's ξ-function on the critical line’, J. Number Theory 16 (1983), 48–74.CrossRefGoogle Scholar
[13]van der Corput, J. G., ‘Sur l'hypothèse de Goldbach’, Proc. Akad. Wet. Amsterdam 41 (1938), 76–80.Google Scholar
[14]Davenport, H., ‘On Waring's problem for fourth powers’, Ann. of Math. 40 (1939), 189–198.CrossRefGoogle Scholar
[15]Davenport, H., ‘On Waring's problem for fifth and sixth powers’, Amer. J. Math. 64 (1942), 199–207.CrossRefGoogle Scholar
[16]Davenport, H., ‘Cubic forms in sixteen variables’, Proc. Royal Soc. London A 272 (1963), 285–303.Google Scholar
[17]Davenport, H. and Heilbronn, H., ‘On Waring's problem: two cubes and one square’, Proc. London Math. Soc. 43 (1937), 73–104.Google Scholar
[18]Davenport, H. and Heilbronn, H., ‘Note on a result in the additive theory of numbers’, Proc. London Math. Soc. 43 (1937), 142–151.Google Scholar
[19]Davenport, H. and Heilbronn, H., ‘On indefinite quadratic forms in five variables’, J. London Math. Soc. 21 (1946), 185–193.CrossRefGoogle Scholar
[20]Davidson, M., ‘On Waring's problem in number fields’, J. London Math. Soc. (to appear).Google Scholar
[21]Davidson, M., ‘On Siegel's conjecture in waring's problem’, Mathematika (to appear).Google Scholar
[22]Davidson, M., ‘Sums of k-th powers in number fields’, Mathematika (to appear).Google Scholar
[23]Effinger, G. W. and Hayes, D. R., Additive number theory of polynomials over a finite field, Oxford Mathematical Monographs (Clarendon Press, Oxford, 1991).CrossRefGoogle Scholar
[24]Elliott, P. D. T. A., Probabilistic number theory: mean value theorems, Grundlehren der Math. Wiss. 239 (Springer-Verlag, New York, Berlin, Heidelberg, 1979).CrossRefGoogle Scholar
[25]Elliott, P. D. T. A., Probabilistic number theory: central limit theorems, Grundlehren der Math. Wiss. 240 (Springer-Verlag, New York, Berlin, Heidelberg, 1979).CrossRefGoogle Scholar
[26]Elliott, P. D. T. A., Duality in analytic number theory (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
[27]Elliott, P. D. T. A. and Halberstam, H., ‘Some applications of Bombieri's theorem’, Mathematika 13 (1966), 196–203.CrossRefGoogle Scholar
[28]Erdős, P., ‘The difference of consecutive primes’, Duke Math. J. 6 (1940), 438–441.CrossRefGoogle Scholar
[29]Erdős, P. and Kac, M., ‘On the Gaussian law of errors in the theory of additive functions’, Proc. Nat. Acad. Sci. USA 25 (1939), 206–207.CrossRefGoogle ScholarPubMed
[30]Erdős, P. and Kac, M., ‘On the Gaussian law of errors in the theory of additive number theoretic functions’, Amer. J. Math. 62 (1940), 738–742.CrossRefGoogle Scholar
[31]Estermann, T., ‘On Goldbach's problem: Proof that almost all even positive integers are sums of two primes’, Proc. London Math. Soc. 44 (1938), 307–314.CrossRefGoogle Scholar
[32]Ford, K. B., ‘New estimates for mean values of Weyl sums’, in: Internat. Math. Res. Notices (1995) pp. 155–171.CrossRefGoogle Scholar
[33]Furstenberg, H., ‘Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions’, J. d'Analyse Math. 31 (1977), 204–256.CrossRefGoogle Scholar
[34]Gallagher, P. X., ‘The large sieve’, Mathematika 14 (1966), 14–20.CrossRefGoogle Scholar
[35]Goldston, D. A. and Vaughan, R. C., ‘On the Montgomery–Hooley asymptotic formula’, in: Sieve Methods, Exponential sums and their Applications in Number Theory (eds. Greaves, G. R. H., Harman, G. and Huxley, M. N.) (Cambridge University Press, Cambridge, 1996).Google Scholar
[36]Hardy, G. H., A Course of Pure Mathematics (Cambridge University Press, Cambridge, 1908).Google Scholar
[37]Hardy, G. H., ‘Sur les zéros de la fonction ζ(s) de Riemann’, Comptes Rendus Acad. Sci. Paris Sér A 158 (1914), 1012–1014.Google Scholar
[38]Hardy, G. H., ‘Goldbach's theorem’, Math. Tid. B (1922), 1–16.Google Scholar
[39]Hardy, G. H., A Mathematician's Apology (Cambridge University Press, Cambridge, 1940).Google Scholar
[40]Hardy, G. H., Ramanujan. Twelve lectures on subjects suggested by his life and work (Cambridge University Press, Cambridge, 1940).Google Scholar
[41]Hardy, G. H. and Littlewood, J. E., ‘Contributions to the theory of the Riemann zeta–function and the theory of the distribution of primes’, Acta Mathematica 41 (1918), 119–196.CrossRefGoogle Scholar
[42]Hardy, G. H. and Littlewood, J. E., ‘A new solution of Waring's problem’, Quart. J. Math. Oxford 48 (1919). 272–293.Google Scholar
[43]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. I A new solution of Waring's problem’. Göttingen Nachrichten (1920), 33–54.Google Scholar
[44]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. II Proof that every large number is the sum of at most 21 biquadrates’, Math. Z. 9 (1921), 14–27.CrossRefGoogle Scholar
[45]Hardy, G. H. and Littlewood, J. E., ‘The zeros of Riemann's zeta–function on the critical line’, Math. Z. 10 (1921), 283–317.CrossRefGoogle Scholar
[46]Hardy, G. H. and Littlewood, J. E., ‘The approximate functional equation in the theory of the zeta function, with applications to the divisor problems of Dirichlet and Piltz’, Proc. London Math. Soc. 21 (1922), 39–74.Google Scholar
[47]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. IV The singular series in Waring's problem’, Math. Z. 12 (1922), 161–188.CrossRefGoogle Scholar
[48]Hardy, G. H. and Littlewood, J. E., ‘On Lindelöf's hypothesis concerning the Riemann zeta–function’, Proc. Roy. Soc. A 103 (1923), 403–412.Google Scholar
[49]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. III On the expression of a number as a sum primes’ Acta Math. 44 (1923), 1–70.CrossRefGoogle Scholar
[50]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. V A further contribution to the study of Goldbach's problem’, Proc. London Math. Soc. 22 (1923), 46–56.Google Scholar
[51]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”. VI Further researches in Waring's problem’, Math. Z. 23 (1925), 1–37.CrossRefGoogle Scholar
[52]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio Numerorum”, VIII The number γ(k) in Warning's problem’, Proc. London Math. Soc. 28 (1928), 518–541.CrossRefGoogle Scholar
[53]Hardy, G. H. and Littlewood, J. E., ‘The approximate functional equation for ζ(s) and ζ(s 2)’ Proc. London Math. Soc. 29 (1929), 81–97.CrossRefGoogle Scholar
[54]Hardy, G. H., Littlewood, J. E. and Pòlya, G., Inequalities second edition (Cambridge University Press, Cambridge, 1959).Google Scholar
[55]Hardy, G. H. and Ramanujan, S., ‘Asymptotic formulae in combinatory analysis’, Proc. London. Math. Soc. 17 (1918), 75–115.CrossRefGoogle Scholar
[56]Hardy, G. H. and Ramanujan, S., The normal number of prime factors of a number n’, Quart. J. Math. Oxford 48 (1920), 76–92.Google Scholar
[57]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1938).Google Scholar
[58]Harman, G., ‘Diophantine approximation by prime numbers’, J. London Math. Soc. 44 (1991), 218–226.CrossRefGoogle Scholar
[59]Heath-Brown, D. R., ‘Cubic forms in ten variables’, Proc. London Math. Soc. 47 (1983), 225–257.CrossRefGoogle Scholar
[60]Heath-Brown, D. R., ‘Weyl's inequality, Hua's inequality, and Waring's problem’, J. London Math. Soc. 23 (1988), 396–414.Google Scholar
[61]Heath-Brown, D. R., ‘Weyl's inequality, Hua's inequality’, in: Number Theory (Ulm, 1987), Lecture Notes in Math. 1380 (Springer-Verlag, Berlin, 1989) pp. 67–92.Google Scholar
[62]Heath-Brown, D. R., ‘The density of zeros of forms for which weak approximation fails’, Math. Comp. 59 (1992), 613–623.CrossRefGoogle Scholar
[63]Hilbert, D., ‘Beweis für Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl unter Potenzen Waringsche Problem’, Math. Annalen 67 (1909), 281–300;CrossRefGoogle Scholar
Nachrichten von der Königlichen Gesellchaft der Wissenschaften zu Göttingen mathematischphysikalische Klasse aus den Jahre 1909, pp. 17–36;.Google Scholar
[64]Hooley, C., ‘On the representation of a number as the sum of two squares and a prime’, Acta Math. 97 (1957), 189–210.CrossRefGoogle Scholar
[65]Hooley, C., ‘On nonary cubic forms. I’, J. reine angew. Math. 386 (1988), 32–98.Google Scholar
[66]Hooley, C., ‘On nonary cubic forms. II’, J. reine angew. Math. 415 (1991), 95–165.Google Scholar
[67]Hooley, C., ‘On nonary cubic forms. III’, J. reine angew. Math. 456 (1994), 53–63.Google Scholar
[68]Hua, L.-K., ‘On Waring's problem’, Quart. J. Math. Oxford 9 (1938), 199–202.CrossRefGoogle Scholar
[69]Huxley, M. N., ‘Small differences between consecutive primes, I’, Mathematika 20 (1973), 229–232.CrossRefGoogle Scholar
[70]Huxley, M. N., ‘Small differences between consecutive primes, II’, Mathematika 24 (1977), 142–152.CrossRefGoogle Scholar
[71]Huxley, M. N., Area, Lattice Points, and Exponential Sums, London Math. Soc. Monographs, New series 13 (Clarendon Press, Oxford, 1996).CrossRefGoogle Scholar
[72]Kubilius, J., ‘Probabilistic methods in the theory of numbers’, Uspeki Mat. Nauk 11 (1956), 31–66Google Scholar
American Math. Soc. Transl. 19 (1962) pp. 47–85.Google Scholar
[73]Levinson, N., ‘More than one third of the zeros of Riemann's zeta-function are on σ = ½’, Adv. Math. 13 (1974), 383–436.CrossRefGoogle Scholar
[74]Levinson, N., ‘A simplification of the proof that N 0(T) > ⅓N(T) for Riemann's zeta-function’, Adv. Math. 18 (1975), 239–242.CrossRefGoogle Scholar
[75]Levinson, N., ‘Deduction of semi-optimal mollifier for obtaining lower bounds for N 0(T) for Riemann's zeta-function’, Proc. Nat. Acad. Sci. USA 72 (1975), 294–297.CrossRefGoogle Scholar
[76]Linnik, Ju. V., ‘The large sieve’, Dokl. Akad. Nauk SSSR 30 (1941), 292–294.Google Scholar
[77]Linnik, Ju. V., The dispersion method in binary additive problems Transl. by Schuur, S. (Amer. Math. Soc., Providence, 1963).Google Scholar
[78]Lou, S.-T., ‘A lower bound for the number of zeros of Riemann's zeta–function on σ = ½’, in: Recent Progress in Analytic Number Theory. Vol. I (Academic Press, London, 1981) pp. 319–324.Google Scholar
[79]Mahler, K., ‘On the fractional parts of the powers of a rational number II’, Mathematica 4 (1957), 122–124.Google Scholar
[80]Maier, H., ‘Small differences between prime numbers’, Mich. Math. J. 35 (1988), 323–344.CrossRefGoogle Scholar
[81]Montgomery, H. L., ‘A note on the large service’, J. London Math. Soc. 43 (1968), 93–98.CrossRefGoogle Scholar
[82]Montgomery, H. L., ‘Primes in arithmetic progressions’, Mich. Math. J. 17 (1970), 33–39.CrossRefGoogle Scholar
[83]Montgomery, H. L. and Vaughan, R. C., ‘The large sieve’, Mathematika 20 (1973), 119–13.CrossRefGoogle Scholar
[84]Montgomery, H. L. and Vaughan, R. C., ‘Error terms in additive prime number theory’, Quart. J. Math. Oxford 24 (1973), 207–216.CrossRefGoogle Scholar
[85]Montgomery, H. L. and Vaughan, R. C., ‘Hilbert's inequality’, J. London Math. Soc. 8 (1974), 73–82.CrossRefGoogle Scholar
[86]Montgomery, H. L. and Vaughan, R. C., ‘The exceptional set in Goldbach's problem’, Acta Arithmetica 27 (1975), 353–370.CrossRefGoogle Scholar
[87]Rademacher, H., ‘On the partition function’, London Math. Soc 43 (1937), 241–254.Google Scholar
[88]Rademacher, H., ‘Additive algebraic number theory’, Proc. Intern. Congr. Math. 1 (1950), 356–362.Google Scholar
[89]Ramanujan, S., ‘Highly composite numbers’, Proc. London Math. Soc. 14 (1915), 347–409.CrossRefGoogle Scholar
[90]Rankin, R. A., ‘The difference between consecutive primes’, J. London Math. Soc. 13 (1938), 242–247.CrossRefGoogle Scholar
[91]Rankin, R. A., ‘The difference between consecutive primes, II’, Proc. Cam. Phil. Soc. 36 (1940), 255–266.CrossRefGoogle Scholar
[92]Rényi, A., ‘On the large sieve of Ju. V. Linnik’, Compositio Math. 8 (1950), 68–75.Google Scholar
[93]Ricci, G., ‘Rechereches sur l'allure de la suite ’, in: Colloque sur le Théorie des Nombres Bruxelles 1955 (Georges Thone, Liege; Masson and Cie, Paris, 1956) pp. 93–106.Google Scholar
[94]Roth, K. F., ‘A problem in additive number theory’, Proc. London Math. Soc. 53 (1951), 381–395.CrossRefGoogle Scholar
[95]Roth, K. F., ‘On certain sets of integers I’, J. London Math. Soc. 28 (1953), 104–109.CrossRefGoogle Scholar
[96]Roth, K. F., ‘On certain sets of integers II’, J. London Math. Soc. 29 (1954), 2–26.Google Scholar
[97]Roth, K. F., ‘On the large sieves of Linnik and Renyi’, Mathematika 12 (1965), 1–9.CrossRefGoogle Scholar
[98]Schmidt, W. M., ‘Small zeros of additive forms in many variables I’, Trans. Amer. Math. Soc. 248 (1979), 121–133.CrossRefGoogle Scholar
[99]Schmidt, W. M., ‘Small zeros of additive forms in many variables II’, Acta Math. 143 (1979), 219–232.CrossRefGoogle Scholar
[100]Schmidt, W. M., ‘Diophantine inequalities for forms of odd degrees’, Advances in Math. 38 (1980), 128–151.CrossRefGoogle Scholar
[101]Schmidt, W. M., ‘The density of integer points on homogeneous varieties’, Acta Math. 154 (1985), 243–296.CrossRefGoogle Scholar
[102]Selberg, A., ‘On the zeros of Riemann's zeta–function on the critical line’, Skr. Norske Vid. Akad. Oslo 10 (1942), ????Google Scholar
[103]Selberg, A., ‘Reflections around the Ramanujan centenary’, in: Atle Selberg Collected Papers, Vol. I (Springer-Verlag, Berlin Heidelberg, 1989) pp. 695–706.Google Scholar
[104]Selberg, A., ‘Lectures on sieves’, in: Atle Selberg Collected Papers, Vol. II (Springer-Verlag, Berlin Heidelberg, 1991) pp. 65–247.Google Scholar
[105]Siegel, C. L., ‘Generalization of Waring's problem to algebraic number fields’, Am. J. Math. 66 (1944), 122–136.CrossRefGoogle Scholar
[106]Siegel, C. L., ‘Sums of mth powers of algebraic integers’, Ann. Math. 246 (1945), 313–339.CrossRefGoogle Scholar
[107]Szemetédi, E., ‘On sets of integers containing no K elements in arithmetic progression’, Acta Arithmetica 27 (1975), 199–245.CrossRefGoogle Scholar
[108]Tenenbaum, G., Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics: 46 (Cambridge University Press, Cambridge, 1995).Google Scholar
[109]Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (Oxford University Press, Oxford, 1951).Google Scholar
[110]Turán, P., ‘On a theorem of Hardy and Ramanujan’, J. London Math. Soc. 9 (1934), 274–276.CrossRefGoogle Scholar
[111]Vaughan, R. C., ‘Diophantine approximation by prime numbers I, II’, Proc. London Math. Soc. 28 (1974), 373–384; 385–401.CrossRefGoogle Scholar
[112]Vaughan, R. C., ‘Sommes trigonométriques sur les nombres premiers’, C. R. Acad. Sci. Paris Sér. A 258 (1977), 981–983.Google Scholar
[113]Vaughan, R. C., ‘An elementary method in prime number theory’, Acta Arithmetica 37 (1980), 111–115.CrossRefGoogle Scholar
[114]Vaughan, R. C., ‘A temary additive problem’, Proc. London Math. Soc. 41 (1980), 516–532.CrossRefGoogle Scholar
[115]Vaughan, R. C., ‘On Waring's problem for cubes’, J. reine angew. Math. 365 (1986), 122–170.Google Scholar
[116]Vaughan, R. C., ‘On Waring's problem for smaller exponents. II’, Mathematika 33 (1986), 6–22.CrossRefGoogle Scholar
[117]Vaughan, R. C., ‘A new iterative method in Waring's problem’, Acta Math. 162 (1989), 1–71.CrossRefGoogle Scholar
[118]Vaughan, R. C., ‘On a variance associated with the distribution of general sequences in arithmetic progressions I, II’, Phil. Trans. Royal Soc. London A 356 (1998), 781–791.CrossRefGoogle Scholar
[119]Vaughan, R. C. and Wooley, T. D., ‘Further improvements in Waring's problem. III: Eighth powers’, Phil. Trans. Royal Soc. London A 354 (1993), 385–396.Google Scholar
[120]Vaughan, R. C. and Wooley, T. D., ‘Further improvements in Waring's problem. II: Sixth powers’, Duke Math. J.? (1994), 683–710.Google Scholar
[121]Vaughan, R. C. and Wooley, T. D., ‘Further improvements in Waring's problem’, Acta Math. 174 (1995), 147–240.CrossRefGoogle Scholar
[122]Vinogradov, I. M., ‘Some theorems concerning the theory of primes’, Recueil Math. 44 (1937), 179–195.Google Scholar
[123]Vinogradov, I. M., ‘On an upper bound for G(n)’, Lzv. Akad. Nauk SSSR 23 (1959), 637–642.Google Scholar
[124]Waring, E., Meditations Algebraicœ English translation of the third edition, 1782 (American Math. Soc., Providence, 1991).Google Scholar
[125]Weyl, H., ‘Über die Geichverteilung von Zahlen mod Eins’, Math. Ann. 77 (1919), 313–352.CrossRefGoogle Scholar
[126]Wooley, T. D., ‘Large improvements in Waring's problem’, Ann. Math. 162 (1992), 1–71.Google Scholar
[127]Wooley, T. D., ‘On Vinogradov's mean value theorem’, Mathematika 39 (1993), 379–399.CrossRefGoogle Scholar