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Hardy's legacy to number theory

  • R. C. Vaughan (a1)
Abstract
Abstract

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.

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References
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[1]Baker R. C. and Harman G., ‘Diophantine approximation by prime numbers’, J. London Math. Soc. 25 (1982), 201215.
[2]Birch B. J., ‘Waring's problem in algebraic number fields’, Proc. Cam. Phil. Soc. 57 (1961), 449459.
[3]Birch B. J., ‘Forms in many variables’, Proc. Royal Soc. London Ser. A 265 (1962), 245263.
[4]Boklan K. D., ‘The asymptotic formula in Waring's problem’, Mathematika 41 (1994), 147161.
[5]Bombieri E., ‘On the large sieve’, Mathematika 12 (1965), 201225.
[6]Bombieri E. and Davenport H., ‘Small differences between prime numbers’, Proc. Royal Soc. London Ser. A 293 (1966), 118.
[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. 1122.
[8]Bombieri E. and Iwaniec H., ‘On the order ζ(1/2+it)’, Ann. Scuola Norm. Pisa CI. Sci. 13 (1986), 449472.
[9]Bombieri E. and Iwaniec H., ‘Some mean value theorems for exponential sums’, Ann. Scuola Norm. Pisa Cl. Sci. 13 (1986), 473486.
[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), 385386 Foreign Lang. Ed.
[11]Chudakov N. G., ‘On the Goldbach problem’, Comptes Rendus Acad. Sci. URSS 17 (1937), 335338.
[12]Conrey J. B., ‘Zeros of derivatives of Riemann's ξ-function on the critical line’, J. Number Theory 16 (1983), 4874.
[13]van der Corput J. G., ‘Sur l'hypothèse de Goldbach’, Proc. Akad. Wet. Amsterdam 41 (1938), 7680.
[14]Davenport H., ‘On Waring's problem for fourth powers’, Ann. of Math. 40 (1939), 189198.
[15]Davenport H., ‘On Waring's problem for fifth and sixth powers’, Amer. J. Math. 64 (1942), 199207.
[16]Davenport H., ‘Cubic forms in sixteen variables’, Proc. Royal Soc. London A 272 (1963), 285303.
[17]Davenport H. and Heilbronn H., ‘On Waring's problem: two cubes and one square’, Proc. London Math. Soc. 43 (1937), 73104.
[18]Davenport H. and Heilbronn H., ‘Note on a result in the additive theory of numbers’, Proc. London Math. Soc. 43 (1937), 142151.
[19]Davenport H. and Heilbronn H., ‘On indefinite quadratic forms in five variables’, J. London Math. Soc. 21 (1946), 185193.
[20]Davidson M., ‘On Waring's problem in number fields’, J. London Math. Soc. (to appear).
[21]Davidson M., ‘On Siegel's conjecture in waring's problem’, Mathematika (to appear).
[22]Davidson M., ‘Sums of k-th powers in number fields’, Mathematika (to appear).
[23]Effinger G. W. and Hayes D. R., Additive number theory of polynomials over a finite field, Oxford Mathematical Monographs (Clarendon Press, Oxford, 1991).
[24]Elliott P. D. T. A., Probabilistic number theory: mean value theorems, Grundlehren der Math. Wiss. 239 (Springer-Verlag, New York, Berlin, Heidelberg, 1979).
[25]Elliott P. D. T. A., Probabilistic number theory: central limit theorems, Grundlehren der Math. Wiss. 240 (Springer-Verlag, New York, Berlin, Heidelberg, 1979).
[26]Elliott P. D. T. A., Duality in analytic number theory (Cambridge University Press, Cambridge, 1997).
[27]Elliott P. D. T. A. and Halberstam H., ‘Some applications of Bombieri's theorem’, Mathematika 13 (1966), 196203.
[28]Erdős P., ‘The difference of consecutive primes’, Duke Math. J. 6 (1940), 438441.
[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), 206207.
[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), 738742.
[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), 307314.
[32]Ford K. B., ‘New estimates for mean values of Weyl sums’, in: Internat. Math. Res. Notices (1995) pp. 155171.
[33]Furstenberg H., ‘Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions’, J. d'Analyse Math. 31 (1977), 204256.
[34]Gallagher P. X., ‘The large sieve’, Mathematika 14 (1966), 1420.
[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).
[36]Hardy G. H., A Course of Pure Mathematics (Cambridge University Press, Cambridge, 1908).
[37]Hardy G. H., ‘Sur les zéros de la fonction ζ(s) de Riemann’, Comptes Rendus Acad. Sci. Paris Sér A 158 (1914), 10121014.
[38]Hardy G. H., ‘Goldbach's theorem’, Math. Tid. B (1922), 116.
[39]Hardy G. H., A Mathematician's Apology (Cambridge University Press, Cambridge, 1940).
[40]Hardy G. H., Ramanujan. Twelve lectures on subjects suggested by his life and work (Cambridge University Press, Cambridge, 1940).
[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), 119196.
[42]Hardy G. H. and Littlewood J. E., ‘A new solution of Waring's problem’, Quart. J. Math. Oxford 48 (1919). 272293.
[43]Hardy G. H. and Littlewood J. E., ‘Some problems of “Partitio Numerorum”. I A new solution of Waring's problem’. Göttingen Nachrichten (1920), 3354.
[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), 1427.
[45]Hardy G. H. and Littlewood J. E., ‘The zeros of Riemann's zeta–function on the critical line’, Math. Z. 10 (1921), 283317.
[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), 3974.
[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), 161188.
[48]Hardy G. H. and Littlewood J. E., ‘On Lindelöf's hypothesis concerning the Riemann zeta–function’, Proc. Roy. Soc. A 103 (1923), 403412.
[49]Hardy G. H. and Littlewood J. E., ‘Some problems of “Partitio Numerorum”. III On the expression of a number as a sum primesActa Math. 44 (1923), 170.
[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), 4656.
[51]Hardy G. H. and Littlewood J. E., ‘Some problems of “Partitio Numerorum”. VI Further researches in Waring's problem’, Math. Z. 23 (1925), 137.
[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), 518541.
[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), 8197.
[54]Hardy G. H., Littlewood J. E. and Pòlya G., Inequalities second edition (Cambridge University Press, Cambridge, 1959).
[55]Hardy G. H. and Ramanujan S., ‘Asymptotic formulae in combinatory analysis’, Proc. London. Math. Soc. 17 (1918), 75115.
[56]Hardy G. H. and Ramanujan S., The normal number of prime factors of a number n’, Quart. J. Math. Oxford 48 (1920), 7692.
[57]Hardy G. H. and Wright E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1938).
[58]Harman G., ‘Diophantine approximation by prime numbers’, J. London Math. Soc. 44 (1991), 218226.
[59]Heath-Brown D. R., ‘Cubic forms in ten variables’, Proc. London Math. Soc. 47 (1983), 225257.
[60]Heath-Brown D. R., ‘Weyl's inequality, Hua's inequality, and Waring's problem’, J. London Math. Soc. 23 (1988), 396414.
[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. 6792.
[62]Heath-Brown D. R., ‘The density of zeros of forms for which weak approximation fails’, Math. Comp. 59 (1992), 613623.
[63]Hilbert D., ‘Beweis für Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl unter Potenzen Waringsche Problem’, Math. Annalen 67 (1909), 281300;
Nachrichten von der Königlichen Gesellchaft der Wissenschaften zu Göttingen mathematischphysikalische Klasse aus den Jahre 1909, pp. 1736;.
[64]Hooley C., ‘On the representation of a number as the sum of two squares and a prime’, Acta Math. 97 (1957), 189210.
[65]Hooley C., ‘On nonary cubic forms. I’, J. reine angew. Math. 386 (1988), 3298.
[66]Hooley C., ‘On nonary cubic forms. II’, J. reine angew. Math. 415 (1991), 95165.
[67]Hooley C., ‘On nonary cubic forms. III’, J. reine angew. Math. 456 (1994), 5363.
[68]Hua L.-K., ‘On Waring's problem’, Quart. J. Math. Oxford 9 (1938), 199202.
[69]Huxley M. N., ‘Small differences between consecutive primes, I’, Mathematika 20 (1973), 229232.
[70]Huxley M. N., ‘Small differences between consecutive primes, II’, Mathematika 24 (1977), 142152.
[71]Huxley M. N., Area, Lattice Points, and Exponential Sums, London Math. Soc. Monographs, New series 13 (Clarendon Press, Oxford, 1996).
[72]Kubilius J., ‘Probabilistic methods in the theory of numbers’, Uspeki Mat. Nauk 11 (1956), 3166
American Math. Soc. Transl. 19 (1962) pp. 4785.
[73]Levinson N., ‘More than one third of the zeros of Riemann's zeta-function are on σ = ½’, Adv. Math. 13 (1974), 383436.
[74]Levinson N., ‘A simplification of the proof that N 0(T) > ⅓N(T) for Riemann's zeta-function’, Adv. Math. 18 (1975), 239242.
[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), 294297.
[76]Linnik Ju. V., ‘The large sieve’, Dokl. Akad. Nauk SSSR 30 (1941), 292294.
[77]Linnik Ju. V., The dispersion method in binary additive problems Transl. by Schuur S. (Amer. Math. Soc., Providence, 1963).
[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. 319324.
[79]Mahler K., ‘On the fractional parts of the powers of a rational number II’, Mathematica 4 (1957), 122124.
[80]Maier H., ‘Small differences between prime numbers’, Mich. Math. J. 35 (1988), 323344.
[81]Montgomery H. L., ‘A note on the large service’, J. London Math. Soc. 43 (1968), 9398.
[82]Montgomery H. L., ‘Primes in arithmetic progressions’, Mich. Math. J. 17 (1970), 3339.
[83]Montgomery H. L. and Vaughan R. C., ‘The large sieve’, Mathematika 20 (1973), 119–13.
[84]Montgomery H. L. and Vaughan R. C., ‘Error terms in additive prime number theory’, Quart. J. Math. Oxford 24 (1973), 207216.
[85]Montgomery H. L. and Vaughan R. C., ‘Hilbert's inequality’, J. London Math. Soc. 8 (1974), 7382.
[86]Montgomery H. L. and Vaughan R. C., ‘The exceptional set in Goldbach's problem’, Acta Arithmetica 27 (1975), 353370.
[87]Rademacher H., ‘On the partition function’, London Math. Soc 43 (1937), 241254.
[88]Rademacher H., ‘Additive algebraic number theory’, Proc. Intern. Congr. Math. 1 (1950), 356362.
[89]Ramanujan S., ‘Highly composite numbers’, Proc. London Math. Soc. 14 (1915), 347409.
[90]Rankin R. A., ‘The difference between consecutive primes’, J. London Math. Soc. 13 (1938), 242247.
[91]Rankin R. A., ‘The difference between consecutive primes, II’, Proc. Cam. Phil. Soc. 36 (1940), 255266.
[92]Rényi A., ‘On the large sieve of Ju. V. Linnik’, Compositio Math. 8 (1950), 6875.
[93]Ricci G., ‘Rechereches sur l'allure de la suite inline-graphic’, in: Colloque sur le Théorie des Nombres Bruxelles 1955 (Georges Thone, Liege; Masson and Cie, Paris, 1956) pp. 93106.
[94]Roth K. F., ‘A problem in additive number theory’, Proc. London Math. Soc. 53 (1951), 381395.
[95]Roth K. F., ‘On certain sets of integers I’, J. London Math. Soc. 28 (1953), 104109.
[96]Roth K. F., ‘On certain sets of integers II’, J. London Math. Soc. 29 (1954), 226.
[97]Roth K. F., ‘On the large sieves of Linnik and Renyi’, Mathematika 12 (1965), 19.
[98]Schmidt W. M., ‘Small zeros of additive forms in many variables I’, Trans. Amer. Math. Soc. 248 (1979), 121133.
[99]Schmidt W. M., ‘Small zeros of additive forms in many variables II’, Acta Math. 143 (1979), 219232.
[100]Schmidt W. M., ‘Diophantine inequalities for forms of odd degrees’, Advances in Math. 38 (1980), 128151.
[101]Schmidt W. M., ‘The density of integer points on homogeneous varieties’, Acta Math. 154 (1985), 243296.
[102]Selberg A., ‘On the zeros of Riemann's zeta–function on the critical line’, Skr. Norske Vid. Akad. Oslo 10 (1942), ????
[103]Selberg A., ‘Reflections around the Ramanujan centenary’, in: Atle Selberg Collected Papers, Vol. I (Springer-Verlag, Berlin Heidelberg, 1989) pp. 695706.
[104]Selberg A., ‘Lectures on sieves’, in: Atle Selberg Collected Papers, Vol. II (Springer-Verlag, Berlin Heidelberg, 1991) pp. 65247.
[105]Siegel C. L., ‘Generalization of Waring's problem to algebraic number fields’, Am. J. Math. 66 (1944), 122136.
[106]Siegel C. L., ‘Sums of mth powers of algebraic integers’, Ann. Math. 246 (1945), 313339.
[107]Szemetédi E., ‘On sets of integers containing no K elements in arithmetic progression’, Acta Arithmetica 27 (1975), 199245.
[108]Tenenbaum G., Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics: 46 (Cambridge University Press, Cambridge, 1995).
[109]Titchmarsh E. C., The Theory of the Riemann Zeta-Function (Oxford University Press, Oxford, 1951).
[110]Turán P., ‘On a theorem of Hardy and Ramanujan’, J. London Math. Soc. 9 (1934), 274276.
[111]Vaughan R. C., ‘Diophantine approximation by prime numbers I, II’, Proc. London Math. Soc. 28 (1974), 373384; 385–401.
[112]Vaughan R. C., ‘Sommes trigonométriques sur les nombres premiers’, C. R. Acad. Sci. Paris Sér. A 258 (1977), 981983.
[113]Vaughan R. C., ‘An elementary method in prime number theory’, Acta Arithmetica 37 (1980), 111115.
[114]Vaughan R. C., ‘A temary additive problem’, Proc. London Math. Soc. 41 (1980), 516532.
[115]Vaughan R. C., ‘On Waring's problem for cubes’, J. reine angew. Math. 365 (1986), 122170.
[116]Vaughan R. C., ‘On Waring's problem for smaller exponents. II’, Mathematika 33 (1986), 622.
[117]Vaughan R. C., ‘A new iterative method in Waring's problem’, Acta Math. 162 (1989), 171.
[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), 781791.
[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), 385396.
[120]Vaughan R. C. and Wooley T. D., ‘Further improvements in Waring's problem. II: Sixth powers’, Duke Math. J.? (1994), 683710.
[121]Vaughan R. C. and Wooley T. D., ‘Further improvements in Waring's problem’, Acta Math. 174 (1995), 147240.
[122]Vinogradov I. M., ‘Some theorems concerning the theory of primes’, Recueil Math. 44 (1937), 179195.
[123]Vinogradov I. M., ‘On an upper bound for G(n)’, Lzv. Akad. Nauk SSSR 23 (1959), 637642.
[124]Waring E., Meditations Algebraicœ English translation of the third edition, 1782 (American Math. Soc., Providence, 1991).
[125]Weyl H., ‘Über die Geichverteilung von Zahlen mod Eins’, Math. Ann. 77 (1919), 313352.
[126]Wooley T. D., ‘Large improvements in Waring's problem’, Ann. Math. 162 (1992), 171.
[127]Wooley T. D., ‘On Vinogradov's mean value theorem’, Mathematika 39 (1993), 379399.
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