Book Review
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MathSciNet review:
1567679
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Book Information:
Author:
Jean-Loup Mauclaire
Title:
Intégration et théorie des nombres
Additional book information:
preface by S. Iyanagi. Hermann, Paris, 1986, xi + 152 pp., 160F. ISBN 2-7056-6035-6.
Hédi Daboussi, Caractérisation des fonctions multiplicatives p.p. $B^{\lambda }$ à spectre non vide, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 141–166 (French). MR 597021
Hédi Daboussi, Sur les fonctions multiplicatives ayant une valeur moyenne non nulle, Bull. Soc. Math. France 109 (1981), no. 2, 183–205 (French, with English summary). MR 623788
Hédi Daboussi and Hubert Delange, On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2) 26 (1982), no. 2, 245–264. MR 675168, DOI 10.1112/jlms/s2-26.2.245
Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
Hubert Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. (3) 78 (1961), 273–304 (French). MR 0169829
6. P. G. L. Dirichlet, Über die Bestimmung der mittleren Werthe in der Zahlentheorie, Abh. Akad. Berlin (1849), 69-83; see: Werke, Vol. 2, VI, Reimer, Berlin, 1897.
H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
P. D. T. A. Elliott, A mean-value theorem for multiplicative functions, Proc. London Math. Soc. (3) 31 (1975), no. 4, 418–438. MR 387220, DOI 10.1112/plms/s3-31.4.418
P. D. T. A. Elliott, Mean value theorems for multiplicative functions bounded in mean $\alpha$-power, $\alpha >1$, J. Austral. Math. Soc. Ser. A 29 (1980), no. 2, 177–205. MR 566285
P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
P. D. T. A. Elliott, High-power analogues of the Turán-Kubilius inequality, and an application to number theory, Canadian J. Math. 32 (1980), no. 4, 893–907. MR 590654, DOI 10.4153/CJM-1980-068-0
P. D. T. A. Elliott, Multiplicative functions and Ramanujan’s $\tau$-function, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 4, 461–468. MR 621561
P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 272, Springer-Verlag, New York, 1985. MR 766558, DOI 10.1007/978-1-4613-8548-6
P. D. T. A. Elliott, Functional analysis and additive arithmetic functions, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 2, 179–223. MR 876958, DOI 10.1090/S0273-0979-1987-15507-8
P. D. T. A. Elliott, The norms of compositions of arithmetic operators, Bull. London Math. Soc. 19 (1987), no. 6, 522–530. MR 915428, DOI 10.1112/blms/19.6.522
P. D. T. A. Elliott, C. J. Moreno, and F. Shahidi, On the absolute value of Ramanujan’s $\tau$-function, Math. Ann. 266 (1984), no. 4, 507–511. MR 735531, DOI 10.1007/BF01458543
William John Ellison, Les nombres premiers, Publications de l’Institut de Mathématique de l’Université de Nancago, No. IX, Hermann, Paris, 1975 (French). En collaboration avec Michel Mendès France. MR 0417077
G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 230694, DOI 10.1007/BF01894515
G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745
22. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 Bde., Teubner, Leipzig, 1909.
Carlos J. Moreno and Freydoon Shahidi, The fourth moment of Ramanujan $\tau$-function, Math. Ann. 266 (1983), no. 2, 233–239. MR 724740, DOI 10.1007/BF01458445
Winfried Scharlau and Hans Opolka, From Fermat to Minkowski, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1985. Lectures on the theory of numbers and its historical development; Translated from the German by Walter K. Bühler and Gary Cornell. MR 770936, DOI 10.1007/978-1-4757-1867-6
Karl Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0087685
26. R. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions, Math. Proc. Cambridge Philos. Soc. 35 (1934), 357-372.
R. A. Rankin, Sums of powers of cusp form coefficients. II, Math. Ann. 272 (1985), no. 4, 593–600. MR 807293, DOI 10.1007/BF01455869
Wolfgang Schwarz, Remarks on the theorem of Elliott and Daboussi, and applications, Elementary and analytic theory of numbers (Warsaw, 1982) Banach Center Publ., vol. 17, PWN, Warsaw, 1985, pp. 463–498. MR 840490
Wolfgang Schwarz and Jürgen Spilker, Eine Anwendung des Approximationssatzes von Weierstrass-Stone auf Ramanujan-Summen, Nieuw Arch. Wisk. (3) 19 (1971), 198–209 (German). MR 332704
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
31. E. C. Titchmarsh, The Riemann zeta function, Oxford, 1951; new edition (D. R. Heath-Brown, ed. ), Oxford Univ. Press, 1986.
E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen. II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467 (German). MR 223318, DOI 10.1007/BF02280301
- 1.
- H. Daboussi, Caractérisation des fonctions multiplicatives p. p. B, Ann. Inst. Fourier (Grenoble) 30 (1980), 141-166. MR 0597021
- 2.
- H. Daboussi, Sur les fonctions multiplicatives ayant une valeur moyenne non nulle, Bull. Soc. Math. France 109 (1981), 183-205. MR 623788
- 3.
- H. Daboussi and H. Delange, On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2) 26 (1982), 245-264. MR 675168
- 4.
- H. Davenport, Multiplicative number theory, second edition, revised by H. Montgomery, Graduate Texts in Math., no. 74, Springer-Verlag, Berlin, New York, Heidelberg, 1980. MR 606931
- 5.
- H. Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. 78 (1961), 273-304. MR 169829
- 6.
- P. G. L. Dirichlet, Über die Bestimmung der mittleren Werthe in der Zahlentheorie, Abh. Akad. Berlin (1849), 69-83; see: Werke, Vol. 2, VI, Reimer, Berlin, 1897.
- 7.
- H. M. Edwards, Riemann's zeta function, Academic Press, New York, 1974. MR 466039
- 8.
- P. D. T. A. Elliott, A mean-value theorem for multiplicative functions, Proc. London Math. Soc. 31 (1975), 418-438. MR 387220
- 9.
- P. D. T. A. Elliott, Mean-value theorems for multiplicative functions bounded in mean α-power, α > 1, J. Austral. Math. Soc. Ser. A 29 (1980), 177-205. MR 566285
- 10.
- P. D. T. A. Elliott, Probabilistic number theory. I: Mean-value theorems, II: Central limit theorems, Grundlehren Math. Wiss. 234, 240, Springer-Verlag, Berlin, New York, 1979, 1980. MR 551361
- 11.
- P. D. T. A. Elliott, High-power analogues of the Turán-Kubilius inequality, and an application to number theory, Canad. J. Math. 32 (1980), 893-907. MR 590654
- 12.
- P. D. T. A. Elliott, Multiplicative functions and Ramanujan's τ-function, J. Austral. Math. Soc. Ser. A 30 (1981), 461-468. MR 621561
- 13.
- P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren Math. Wiss., vol. 272, Springer-Verlag, Berlin, New York, Tokyo, 1984/1985. MR 766558
- 14.
- P. D. T. A. Elliott, Functional analysis and additive arithmetic functions, Bull. Amer. Math. Soc. (N. S. ) 16 (1987), 179-223. MR 876958
- 15.
- P. D. T. A. Elliott, The norms of compositions of arithmetic operators, Bull. London Math. Soc. (to appear). MR 915428
- 16.
- P. D. T. A. Elliott, C. Moreno and F. Shahidi, On the absolute value of Ramanujan's τ-function, Math. Ann. 266 (1984), 507-511. MR 735531
- 17.
- W. J. Ellison and M. Mendès-France, Les nombres premiers, Hermann, Paris, 1975. MR 417077
- 18.
- G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365-403. MR 230694
- 19.
- G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge, 1940. MR 4860
- 20.
- A. Ivić, The theory of the Riemann zeta-function with applications, John Wiley and Sons, Ltd., New York, London, Sydney, Toronto, 1985. MR 792089
- 21.
- J. Kubilius, Probabilistic methods in the theory of numbers, Trans. Math. Monographs, vol. 11, Amer. Math. Soc. Providence, R. I., 1964. MR 160745
- 22.
- E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 Bde., Teubner, Leipzig, 1909.
- 23.
- C. Moreno and F. Shahidi, The fourth moment of Ramanujan τ-function, Math. Ann. 266 (1983), 233-239. MR 724740
- 24.
- J. Opolka and W. Scharlau, From Fermat to Minkowski, Springer-Verlag, Berlin, New York, Tokyo, 1985. MR 770936
- 25.
- K. Prachar, Primzahlverteilung, Grundlehren Math. Wiss., 91, Springer-Verlag, Berlin, 1957. MR 87685
- 26.
- R. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions, Math. Proc. Cambridge Philos. Soc. 35 (1934), 357-372.
- 27.
- R. Rankin, Sums of powers of cusp form coefficients. II, Math. Ann. 272 (1985), 593-600. MR 807293
- 28.
- W. Schwarz, Remarks on the theorem of Elliott and Daboussi; and applications, Elementary and Analytic Theory of Numbers (H. Iwaniec, ed. ), Banach Center Publications, Inst, of Math. Polish Acad. Sci., Warszawa, 1985, pp. 463-498. MR 840490
- 29.
- W. Schwarz and J. Spilker, Eine Anwendung des Approximationssatzes von Weierstrass-Stone auf Ramanujan-Summen, Arch. Wisk. (3) 19 (1971), 198-209. MR 332704
- 30.
- F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-355. MR 610479
- 31.
- E. C. Titchmarsh, The Riemann zeta function, Oxford, 1951; new edition (D. R. Heath-Brown, ed. ), Oxford Univ. Press, 1986.
- 32.
- E. Wirsing, Dos asymptotische Verhalten von Summen über multiplikative Funktionen. II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411-467. MR 223318
Review Information:
Reviewer:
P. D. T. A. Elliott
Journal:
Bull. Amer. Math. Soc.
18 (1988), 193-209
DOI:
https://doi.org/10.1090/S0273-0979-1988-15648-0
