Bulletin of the American Mathematical Society

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Book Information:

Author: P. D. T. A. Elliott
Title: Arithmetic functions and integer products
Additional book information: Grundlehren der Mathematischen Wissenschaften, vol. 272, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1985, xv + 461 pp., $64.00. ISBN 0-387-96094-5.

Author: Paul J. McCarthy
Title: Introduction to arithmetical functions
Additional book information: Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1986, vi+365 pp., $35.50. ISBN 0-387-96262-X.

Tom M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics, No. 41, Springer-Verlag, New York-Heidelberg, 1976. MR 0422157

Raymond Ayoub, An introduction to the analytic theory of numbers, Mathematical Surveys, No. 10, American Mathematical Society, Providence, R.I., 1963. MR 0160743

B. J. Birch, Multiplicative functions with non-decreasing normal order, J. London Math. Soc. 42 (1967), 149–151. MR 202675, DOI 10.1112/jlms/s1-42.1.149

E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 197425, DOI 10.1112/S0025579300005313

5.

J. -M. De Koninck and A. Mercier, Approche élémentaire de l'étude des fonctions arithmétiques, Presses Univ. Laval, Québec, 1982.

Hubert Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. (3) 78 (1961), 273–304 (French). MR 0169829

François Dress and Bodo Volkmann, Ensembles d’unicité pour les fonctions arithmétiques additives ou multiplicatives, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 2, A43–A46 (French, with English summary). MR 491439

P. D. T. A. Elliott, A conjecture of Kátai, Acta Arith. 26 (1974/75), 11–20. MR 354599, DOI 10.4064/aa-26-1-11-20

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, 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, 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. Erdös, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1–20. MR 15424, DOI 10.2307/1969031

P. Erdös, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 374–384. MR 29411, DOI 10.1073/pnas.35.7.374

14.

P. Erdős and M. Kac, On the Gaussian law of errors in the theory of additive functions, Proc. Nat. Acad. U. S. A. 25 (1939), 206-207.

P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742. MR 2374, DOI 10.2307/2371483

P. Erdös and L. Mirsky, The distribution of values of the divisor function $d(n)$, Proc. London Math. Soc. (3) 2 (1952), 257–271. MR 49932, DOI 10.1112/plms/s3-2.1.257

17.

J. Galambos and I. Kátai, A new proof for the continuity of the limiting distribution of some arithmetical functions, Proc. Budapest Conference (to appear).

Janos Galambos and Peter Szüsz, On the distribution of multiplicative arithmetical functions, Acta Arith. 47 (1986), no. 1, 57–62. MR 866902, DOI 10.4064/aa-47-1-57-62

Emil Grosswald, Oscillation theorems of arithmetical functions, Trans. Amer. Math. Soc. 126 (1967), 1–28. MR 202685, DOI 10.1090/S0002-9947-1967-0202685-7

G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 230694, DOI 10.1007/BF01894515

H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730

22.

G. Hardy and J. E. Littlewood, Some problems of partitio numerorum. III: On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1-70.

23.

G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. (Oxford) 48 (1917), 76-92.

24.

G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75-115.

D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), no. 1, 141–149. MR 762186, DOI 10.1112/S0025579300010743

Adolf Hildebrand, An asymptotic formula for the variance of an additive function, Math. Z. 183 (1983), no. 2, 145–170. MR 704101, DOI 10.1007/BF01214818

Karl-Heinz Indlekofer, On sets characterizing additive arithmetical functions, Math. Z. 146 (1976), no. 3, 285–290. MR 389736, DOI 10.1007/BF01181885

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

Mark Kac, Statistical independence in probability, analysis and number theory. , The Carus Mathematical Monographs, No. 12, Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1959. MR 0110114

I. Kátai, On sets characterizing number-theoretical functions, Acta Arith. 13 (1967/68), 315–320. MR 220690, DOI 10.4064/aa-13-3-315-320

I. Kátai, Some results and problems in the theory of additive functions, Acta Sci. Math. (Szeged) 30 (1969), 305–311. MR 252339

I. Kátai, On number-theoretical functions, Number Theory (Colloq., János Bolyai Math. Soc., Debrecen, 1968) North-Holland, Amsterdam, 1970, pp. 133–137. MR 0272732

I. Kátai, On a problem of P. Erdős, J. Number Theory 2 (1970), 1–6. MR 250991, DOI 10.1016/0022-314X(70)90002-8

Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), no. 1, 159–182 (French). MR 1554926, DOI 10.1007/BF02403071

G. Kolesnik, On the order of $\zeta ({1\over 2}+it)$ and $\Delta (R)$, Pacific J. Math. 98 (1982), no. 1, 107–122. MR 644943

J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745

L. Moser and J. Lambek, On monotone multiplicative functions, Proc. Amer. Math. Soc. 4 (1953), 544–545. MR 56630, DOI 10.1090/S0002-9939-1953-0056630-3

Jean-Loup Mauclaire, Sur la régularité des fonctions additives, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A431–A433 (French). MR 319872

A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. Reine Angew. Math. 357 (1985), 138–160. MR 783538, DOI 10.1515/crll.1985.357.138

40.

B. Riemann, Über die Anzahl der Primzahlen unter eine gegebenen Grosse, Monatsh. Preuss. Akad. Wiss. (Berlin) (1859), 671-680.

R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), no. 2, 120–126. MR 700103, DOI 10.1145/359340.359342

Atle Selberg, An elementary proof of the prime-number theorem, Ann. of Math. (2) 50 (1949), 305–313. MR 29410, DOI 10.2307/1969455

43.

C. Spiro, Ph. D. Thesis, Univ. of Illinois, Urbana, Ill., 1981.

44.

E C. Titchmarsh, The theory of functions, Oxford University, 1952.

45.

P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274-276.

Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989

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

Eduard Wirsing, Characterisation of the logarithm as an additive function, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 375–381. MR 0412129

E. Wirsing, Additive functions with restricted growth on the numbers of the form $p+1$, Acta Arith. 37 (1980), 345–357. MR 598887, DOI 10.4064/aa-37-1-345-357

Dieter Wolke, Bemerkungen über Eindeutigkeitsmengen additiver Funktionen, Elem. Math. 33 (1978), no. 1, 14–16 (German). MR 463096

1.

T. M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Math., No. 41, Springer-Verlag, 1976. MR 0422157

2.

R. Ayoub, An introduction to the analytic theory of numbers, Amer. Math. Soc. Providence, R.I., 1963. MR 160743

3.

B. J. Birch, Multiplicative functions with non-decreasing normal order, J. London Math. Soc. 42 (1967), 149-151. MR 202675

4.

E. Bombieri, On the large sieve, Mathematika 12 (1965), 201-225. MR 197425

5.

J. -M. De Koninck and A. Mercier, Approche élémentaire de l'étude des fonctions arithmétiques, Presses Univ. Laval, Québec, 1982.

6.

H. Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. 78 (1961), 273-304. MR 169829

7.

F. Dress and B. Volkmann, Ensembles d'unicité pour les fonctions arithmétiques additives ou multiplicatives, C. R. Acad. Sci. Paris Ser. A 287 (1978), 43-46. MR 491439

8.

P. D. T. A. Elliott, A conjecture of Kátai, Acta Arith. 26 (1974), 11-20. MR 354599

9.

P. D. T. A. Elliott, Probabilistic number theory. I, Springer-Verlag, Berlin and New York, 1979. MR 551361

10.

P. D. T. A. Elliott, Probabilistic number theory. II, Springer-Verlag, Berlin and New York, 1980. MR 560507

11.

P. D. T. A. Elliott, Functional analysis and additive arithmetic functions, Bull Amer. Math. Soc. (N.S.) 16 (1987), 179-223. MR 876958

12.

P. Erdős, On the distribution of additive functions, Ann. of Math. (2) 47 (1946), 1-20. MR 15424

13.

P. Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat. Acad. Sci. (Washington) 35 (1949), 374-384. MR 29411

14.

P. Erdős and M. Kac, On the Gaussian law of errors in the theory of additive functions, Proc. Nat. Acad. U. S. A. 25 (1939), 206-207.

15.

P. Erdős and M. Kac, The Gaussian law of errors in the theory of additive number-theoretic functions, Amer. J. Math. 62 (1940), 738-742. MR 2374

16.

P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), 257-271. MR 49932

17.

J. Galambos and I. Kátai, A new proof for the continuity of the limiting distribution of some arithmetical functions, Proc. Budapest Conference (to appear).

18.

J. Galambos and P. Szüsz, On the distribution of multiplicative arithmetical functions, Acta Arith. 47 (1986), 57-62. MR 866902

19.

E. Grosswald, Oscillation theorems of arithmetical functions, Trans. Amer. Math. Soc. 126 (1967), 1-28. MR 202685

20.

G. Halasz, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Sci. Hungar. 19 (1968), 365-403. MR 230694

21.

H. Halberstam and H. E. Richert, Sieve methods, Academic Press, New York, 1974. MR 424730

22.

G. Hardy and J. E. Littlewood, Some problems of partitio numerorum. III: On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1-70.

23.

G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. (Oxford) 48 (1917), 76-92.

24.

G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75-115.

25.

D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), 141-149. MR 762186

26.

A. Hildebrand, An asymptotic formula for the variance of an additive function, Math. Z. 183 (1983), 145-149. MR 704101

27.

K. H. Indlekofer, On sets characterizing additive arithmetical functions, Math. Z. 146 (1976), 285-290. MR 389736

28.

A. Ivić, The Riemann zeta function, John Wiley & Sons, New York, 1985. MR 792089

29.

M. Kac, Statistical independence in probability, analysis and number theory, Carus Math. Monographs, No. 12, Math. Assoc. America (distributed by John Wiley and Sons, New York), 1959. MR 110114

30.

I. Kátai, On sets characterizing number-theoretical functions, Acta Arith. 13 (1968), 315-320. MR 220690

31.

I. Kátai, Some results and problems in the theory of additive functions, Acta Sci. Math. Szeged. 30 (1969), 305-311. MR 252339

32.

I. Kátai, On number-theoretical functions, Colloq. Math. Soc. János Bolyai, 2, North-Holland, Amsterdam, 1970, 133-136. MR 272732

33.

I. Kátai, On a problem of P. Erdős, J. Number Theory 2 (1970), 1-6. MR 250991

34.

H. von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), 159-182. MR 1554926

35.

G. Kolesnik, On the order of $\zeta ({1øver 2}+it)$ and $\Delta (R)$, Pacific J. Math. 98 (1982), 107-122. MR 644943

36.

J. Kubilius, Probabilistic methods in the theory of numbers, Trans. Math. Monographs, vol. 11, Amer. Math. Soc. Providence, R.I., 1968. MR 160745

37.

J. Lambek and L. Moser, On monotone multiplicative functions, Proc. Amer. Math. Soc. 4 (1953), 544-545. MR 56630

38.

J. L. Mauclaire, Sur la régularité des fonctions additives, Séminaire Delange-Pisot-Poitou, Théorie des Nombres, Paris 15 (1973/74), no. 23. MR 319872

39.

A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. Reine Angew Math. 357 (1985), 138-160. MR 783538

40.

B. Riemann, Über die Anzahl der Primzahlen unter eine gegebenen Grosse, Monatsh. Preuss. Akad. Wiss. (Berlin) (1859), 671-680.

41.

R. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), 120-126. MR 700103

42.

A. Selberg, An elementary proof of the prime number theorem, Ann. of Math. (2) 50 (1949), 305-313. MR 29410

43.

C. Spiro, Ph. D. Thesis, Univ. of Illinois, Urbana, Ill., 1981.

44.

E C. Titchmarsh, The theory of functions, Oxford University, 1952.

45.

P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274-276.

46.

P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math., Springer-Verlag, 1987. MR 883451

47.

E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen. II, Acta Math. Akad. Sci. Hungar. 18 (1967), 411-467. MR 223318

48.

E. Wirsing, Characterization of the logarithm as an additive function, Proc. Sympos. Pure Math., vol 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 375-381. MR 412129

49.

E. Wirsing, Additive functions with restricted growth on the numbers of the form p + 1, Acta Arith. 37 (1981), 345-357. MR 598887

50.

D. Wolke, Bemerkungen über Eindeutigkeitmengen additiver Funktionen, Elem. Math. 33 (1978), 14-16. MR 463096