Skip to main content

From Ramanujan to Groups of Rationals: A Personal History of Abstract Multiplicative Functions

  • Conference paper
  • First Online:
Analytic Number Theory, Modular Forms and q-Hypergeometric Series (ALLADI60 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 221))

Included in the following conference series:

  • 1274 Accesses

Abstract

The following furnishes details to a lecture that I gave at the March 17–21, 2016, meeting held in Gainesville, Florida, to celebrate the sixtieth birthday of Krishnaswami Alladi.

In celebration of the sixtieth birthday of Krishnaswami Alladi

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. K. Alladi, The Turán-Kubilius inequality for integers without large prime factors. J. Reine Angew. Math. 335, 180–196 (1982)

    MathSciNet  MATH  Google Scholar 

  2. T. Barnet-Lamb, D. Geraghty, M. Harris, R. Taylor, A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. B.J. Birch, Multiplicative functions with non-decreasing normal order. J. Lond. Math. Soc. 42, 149–151 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. R. de la Bretèche, G. Tenenbaum, On the friable Turán-Kubilius inequality, in Analytic and Probabilistic Methods in Number Theory, Proceedings of the Fifth International Conference in Honour of J. Kubilius, Palanga, Lithuania, 4–10 September, 2011 ed. by F. Schweiger, E. Manstavičius. New trends in probability and statistics (TEV, Vilnius, 2012), pp. 111–117

    Google Scholar 

  5. H. Delange, Sur les fonctions arithmétiques multiplicatives. Ann. Sci. École Norm. Sup. (3) 78(1), 273–304 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)

    Article  Google Scholar 

  7. F. Dress, B. Volkmann, Ensembles d’unicité pour les fonctions arithmétiques additives ou multiplicatives. C. R. Acad. Sci. Paris Sér. A-B 287(2), A43–A46 (1978)

    MATH  Google Scholar 

  8. N.D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb{Q}\). Invent. Math. 89(3), 561–567 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. P.D.T.A. Elliott, Multiplicative functions and the sign of Maass form Fourier coefficients, in From Arithmetic to Zeta functions, a volume in memory of Wolfgang Schwarz, ed. by J. Sander, J. Steuding, R. Steuding (Springer, 2016) pp. 109–120

    Google Scholar 

  10. P.D.T.A. Elliott, On inequalities of large sieve type. Acta Arith. 18(1), 405–422 (1971) (Davenport Memorial Volume)

    Google Scholar 

  11. P.D.T.A. Elliott, A conjecture of Kátai. Acta Arith. 26(1), 11–20 (1974/75)

    Google Scholar 

  12. P.D.T.A. Elliott, The law of large numbers for additive arithmetic functions. Math. Proc. Camb. Philos. Soc. 78(1), 33–71 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. P.D.T.A. Elliott, General asymptotic distributions for additive arithmetic functions. Math. Proc. Camb. Philos. Soc. 79(1), 43–54 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. P.D.T.A. Elliott, On a conjecture of Narkiewicz about functions with non-decreasing normal order. Colloq. Math. 36(2), 289–294 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. P.D.T.A. Elliott, On a problem of Hardy and Ramanujan. Mathematika 23(1), 10–17 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. P.D.T.A. Elliott, On the differences of additive arithmetic functions. Mathematika 24(2), 153–165 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  17. P.D.T.A. Elliott, Probabilistic Number Theory I: Mean-Value Theorems, Grundlehren der mathematischen Wissenschaften, vol. 239 (Springer, New York, 1979)

    Google Scholar 

  18. P.D.T.A. Elliott, Sums and differences of additive arithmetic functions in mean square. J. Reine Angew. Math. 309, 21–54 (1979)

    MathSciNet  MATH  Google Scholar 

  19. P.D.T.A. Elliott, Probabilistic Number Theory II: Central Limit Theorems, Grundlehren der mathematischen Wissenschaften, vol. 240 (Springer, New York, 1980)

    Google Scholar 

  20. P.D.T.A. Elliott, Multiplicative functions and Ramanujan’s \(\tau \)-function. J. Aust. Math. Soc. Ser. A 30(4), 461–468 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. P.D.T.A. Elliott, On representing integers as products of integers of a prescribed type. J. Aust. Math. Soc. Ser. A 35(2), 143–161 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. P.D.T.A. Elliott, Arithmetic Functions and Integer Products, Grundlehren der mathematischen Wissenschaften, vol. 272 (Springer, New York, 1985)

    Google Scholar 

  23. P.D.T.A. Elliott, The norms of compositions of arithmetic operators. Bull. Lond. Math. Soc. 19(6), 522–530 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. P.D.T.A. Elliott, Multiplicative functions \(|g| \le 1\) and their convolutions: an overview, Séminaire de Théorie des Nombres, Paris 1987–88, ed. by C. Goldstein. Progress in Mathematics, vol. 81 (Birkhäuser, Boston, 1990), pp. 63–73

    Google Scholar 

  25. P.D.T.A. Elliott, On the correlation of multiplicative and the sum of additive arithmetic functions. Mem. Am. Math. Soc. 112(538), viii\(+\)88 (1994)

    Google Scholar 

  26. P.D.T.A. Elliott, Duality in Analytic Number Theory, Cambridge Tracts in Mathematics, vol. 122 (Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  27. P.D.T.A. Elliott, Cast thy bread upon the waters \(\dots \) A personal view of the mathematician Paul Erdős, Paul Erdős and his mathematics. I, ed. by G. Halász, L. Lovász, M. Simonovits, V.T. Sós, Bolyai Society Mathematical Studies, vol. 11, János Bolyai Mathematical Society (Budapest, 2002), pp. 175–218

    Google Scholar 

  28. P.D.T.A. Elliott, Product representations by rationals, in Number Theoretic Methods: Future Trends, Proceedings of the Second China-Japan Seminar, Iizuka, Japan, March 12–16, 2001, ed. by S. Kanemitsu, C. Jia. Devolopment in Mathematics, vol. 8 (Kluwer Academic Publications, Dordrecht, 2002), pp. 119–150

    Google Scholar 

  29. P.D.T.A. Elliott, The value distribution of additive arithmetic functions on a line. J. Reine Angew. Math. 642, 57–108 (2010)

    MathSciNet  MATH  Google Scholar 

  30. P.D.T.A. Elliott, An abstract central limit theorem for eigenforms, in Analytic and Probabilistic Methods in Number Theory, Proceedings of the Fifth International Conference in Honour of J. Kubilius, Palanga, Lithuania, 4–10 September, 2011, ed. by E. Manstavičius, A. Laurinčikas, G. Stepanauskas (TEV, Vilnius, 2012), pp. 131–142

    Google Scholar 

  31. P.D.T.A. Elliott, A central limit theorem for Ramanujan’s tau function. Ramanujan J. 29(1–3), 145–161 (2012) (Ramanujan’s 125th anniversary special volume)

    Google Scholar 

  32. P.D.T.A. Elliott, Paul Turán and probabilistic number theory, a Personal View, in Number Theory, Analysis, and Combinatorics, Proceedings of the Paul Turán Memorial Conference held August 22–26, 2011 in Budapest, ed. by J. Pintz, A. Biró, K. Győry, G. Harcos, M. Simonovits, J. Szabados, Proceedings in Mathematics (De Gruyter, Berlin/Boston, 2014), pp. 41–62

    Google Scholar 

  33. P.D.T.A. Elliott, Central limit theorems for classical cusp forms. Ramanujan J. 36(1–2), 81–98 (2015). Basil Gordon memorial volume

    Article  MathSciNet  MATH  Google Scholar 

  34. P.D.T.A. Elliott, Corrigendum: central limit theorems for classical cusp forms. Ramanujan J. 36(1–2), 99–102 (2015) (Basil Gordon memorial volume; published online Oct. 2014)

    Google Scholar 

  35. P.D.T.A. Elliott, Jonas Kubilius and probabilistic number theory. some personal reflections. Lith. Math. J. 55(1), 2–24 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. P.D.T.A. Elliott, J. Kish, Harmonic analysis on the positive rationals. Determination of the group generated by the ratios \((an+b)/({A}n+{B})\) (2016), Mathematik (2017), arXiv:1602.03263

  37. P.D.T.A. Elliott, J. Kish, Harmonic analysis on the positive rationals I: basic results. J. Fac. Sci. Univ. Tokyo 23(3) (2016), arXiv:1405.7130

  38. P.D.T.A. Elliott, J. Kish, Harmonic analysis on the positive rationals II: multiplicative functions and maass forms. J. Math. Sci. Univ. Tokyo 23(3) (2016), arXiv:1405.7132

  39. P.D.T.A. Elliott, C.J. Moreno, F. Shahidi, On the absolute value of Ramanujan’s \(\tau \)-function. Math. Ann. 266(4), 507–511 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  40. P. Erdős, On the distribution function of additive functions. Ann. Math. 47(2), 1–20 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Erdős, M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62, 738–742 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  42. B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley Publishing Co., Reading, 1968) (Translated from the 1949 Russian edition and annotated by K.L. Chung)

    Google Scholar 

  43. D. Goldfeld, J. Hundley, Automorphic Representations and \(L\) Functions for the General Linear Group. Volume I, Cambridge Studies in Advanced Mathematics, vol. 129 (Cambridge University Press, New York, 2011)

    Google Scholar 

  44. G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar. 19, 365–403 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  45. G.H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (Cambridge University Press, Cambridge, 1940) (Reprinted, with additional corrections and a commentary by Bruce C. Berndt, AMS Chelsea, Providence, Rhode Island, 1999)

    Google Scholar 

  46. G.H. Hardy, S. Ramanujan, The normal number of prime factors of a number \(n\). Q. J. Math. 48, 76–92 (1917)

    MATH  Google Scholar 

  47. H. Heilbronn, E. Landau, Anwendungen der N. Wienerschen Methode. Math. Z. 37(1), 18–21 (1933) (c.f. The Collected Papers of Hans Arnold Heilbronn, ed. by E.J. Kari, R.A. Smith. Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, 1988)

    Google Scholar 

  48. H. Heilbronn, E. Landau, Bemerkungen zur vorstehenden Arbeit von Herrn Bochner. Math. Z. 37(1), 10–16 (1933) (c.f. The Collected Papers of Hans Arnold Heilbronn, ed. by E.J. Kari, R.A. Smith. Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, 1988)

    Google Scholar 

  49. A. Hildebrand, Multiplicative functions in short intervals. Can. J. Math. 39(3), 646–672 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Hildebrand, An Erdős-Wintner theorem for differences of additive functions. Trans. Am. Math. Soc. 310(1), 257–276 (1988)

    MATH  Google Scholar 

  51. M. Kac, Note on the distribution of values of the arithmetic function \(d(m)\). Bull. Am. Math. Soc. 47, 815–817 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  52. I. Kátai, On sets characterizing number-theoretical functions. Acta Arith. 13(3), 315–320 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  53. I. Kátai, On sets characterizing number-theoretical functions (II) (The set of “prime plus one” ’s is a set of quasi-uniqueness). Acta Arith. 16(1), 1–4 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  54. I. Kátai, Some results and problems in the theory of additive functions. Acta Sci. Math. (Szeged) 30(3–4), 305–311 (1969)

    MathSciNet  MATH  Google Scholar 

  55. I. Kátai, On number-theoretical functions, in Number Theory, Proceedings of the Number-Theoretic Colloquium, Debrecen, Hungary, April 4–8, 1968, ed. by P. Turán. Colloquia Mathematica Societatis János Bolyai, vol. 2 (North-Holland, Amsterdam, 1970), pp. 133–137

    Google Scholar 

  56. H.H. Kim, F. Shahidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112(1), 177–197 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Kubilius, Probabilistic methods in the theory of numbers (Russian). Uspekhi Mat. Nauk 11(2(68)), 31–66 (1956) (see Am. Math. Soc. Trans. 19, 1962, 47–85)

    Google Scholar 

  58. J. Kubilius, Probabilistic Methods in the Theory of Numbers, 2nd edn. (Vilnius, 1962) (corrected and revised by J. Kubilius, translated from the Russian by G. Burgie and S. Schuur, American Mathematical Society. Translations of Mathematical Monographs, vol. 11, Providence, RI, 1964)

    Google Scholar 

  59. D.H. Lehmer, Ramanujan’s function \(\tau (n)\). Duke Math. J. 10, 483–492 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  60. J. Meyer, Ensembles d’unicité pour les fonctions additives. Étude analogue dans le cas des fonctions multiplicatives, in Proceedings of the Journées de Théorie Analytique et Elémentaire des Nombres, Université de Paris-Sud, Orsay, France, June 2–3, 1980, vol. 81 (Publ. Math. Orsay, no. 1, Université de Paris-Sud, Orsay, 1981), pp. 19–29

    Google Scholar 

  61. L.J. Mordell, On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Camb. Philos. Soc. 19, 117–124 (1917)

    MATH  Google Scholar 

  62. C.J. Moreno, F. Shahidi, The fourth moment of Ramanujan \(\tau \)-function. Math. Ann. 266(2), 233–239 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  63. S. Ramanujan, On certain arithmetical functions. Trans. Camb. Philos. Soc. 22(9), 159–184 (1916)

    Google Scholar 

  64. S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc. 22(13), 259–276 (1918)

    Google Scholar 

  65. S. Ramanujan, in Collected Papers of Srinivasa Ramanujan, ed. by G.H. Hardy, P.V. Seshu Aiyar, B.M. Wilson (Cambridge University Press, Cambridge, 1927) (Reprinted, with a new preface and commentary by Bruce C. Berndt, AMS Chelsea, Providence, Rhode Island, 2000)

    Google Scholar 

  66. R.A. Rankin, Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions I. The zeros of the function \(\sum _{n=1}^\infty \frac{\tau (n)}{n^s}\) on the line \(Re\, s = \frac{13}{2}\). Math. Proc. Camb. Philos. Soc. 35(3), 351–356 (1939)

    Article  MATH  Google Scholar 

  67. R.A. Rankin, Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions II. The order of the Fourier coefficients of integral modular forms. Math. Proc. Camb. Philos. Soc. 35(3), 357–372 (1939)

    Article  MATH  Google Scholar 

  68. R.A. Rankin, Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions III. A note on the sum function of the Fourier coefficients of integral modular forms. Math. Proc. Camb. Philos. Soc. 36(2), 150–151 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  69. R.A. Rankin, Sums of powers of cusp form coefficients. Math. Ann. 263(2), 227–236 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  70. R.A. Rankin, Sums of powers of cusp form coefficients. II. Math. Ann. 272(4), 593–600 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  71. R.A. Rankin, Fourier coefficients of cusp forms. Math. Proc. Camb. Philos. Soc. 100(1), 5–29 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  72. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20, 47–87 (1956)

    MathSciNet  MATH  Google Scholar 

  73. J.-P. Serre, Abelian \(\ell \)-adic representations and elliptic curves, (W.A. Benjamin Inc., New York 1968) (republished as Research Notes in Mathematics, vol. 7, A K Peters Ltd., Wellesley, MA, 1998)

    Google Scholar 

  74. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math. 54, 123–201 (1981)

    Article  MATH  Google Scholar 

  75. F. Shahidi, On certain \(L\)-functions. Am. J. Math. 103(2), 297–355 (1981)

    Article  MATH  Google Scholar 

  76. F. Shahidi, in Symmetric power \({L}\) -functions for \({GL}(2)\), Elliptic curves and related topics (H. Kisilevsky and M. R. Murty, eds.), Centre de Recherches Mathématiques, Université de Montréal, CRM Proceedings and Lecture Notes, vol. 4 (AMS, Providence, RI, 1994), pp. 159–182

    Google Scholar 

  77. T. Tao, The logarithmic averaged Chowla and Elliott conjectures for two-point correlations. Forum. Math. Pi. 4(e8), 36 (2016), arXiv:1509.05422v2

  78. J. Tate, Algebraic cycles and poles of zeta functions, in Arithmetical Algebraic Geometry: Proceedings of a Conference Held at Purdue University December 5–7, 1963, ed. by O.F.G. Schilling. Harper’s Series in Modern Mathematics (Harper and Row, New York, 1965), pp. 93–110

    Google Scholar 

  79. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, vol. 13 (Institut Elie Cartan, Université de Nancy I, 1990), English translation Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, Cambridge, 1994) (Second edition of French edition published by SMF, 1996)

    Google Scholar 

  80. P. Turán, On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9(4), 274–276 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  81. P. Turán, Uber einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan. J. Lond. Math. Soc. 11(2), 125–133 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  82. E. Wirsing, A characterization of \(\log n\) as an additive arithmetic function, in Symposia Mathematica, Proceedings of the Rome Conference on Number Theory, Istituto Nazionale di Alta Matematica Roma, December 9–12, 1968 (London), vol. 4 (Academic Press, London, 1970), pp. 45–57

    Google Scholar 

  83. D. Wolke, Bemerkungen über Eindeutigkeitsmengen additiver Funktionen. Elem. Math. 33(1), 14–16 (1978)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. D. T. A. Elliott .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Elliott, P.D.T.A. (2017). From Ramanujan to Groups of Rationals: A Personal History of Abstract Multiplicative Functions. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_16

Download citation

Publish with us

Policies and ethics