Classical theory of arithmetic functions.

*(English)*Zbl 0657.10001This book gives a survey of the algebraic part of the theory of arithmetical functions. The author collected in 17 chapters a large material dealing with the general theory as well as with many particular functions.

The first part (chapters I-VI) starts with introducing four principal convolution products (Dirichlet, Cauchy, Lucas and the unitary product), brings their main properties and then studies various divisor functions, Euler’s totient and the Moebius function, as well as several of their generalizations. Chapter VII introduces multiplicative functions in two variables and in Chapter VIII certain such functions associated with Abelian groups are studied. The next two chapters deal with Ramanujan sums and cyclotomic polynomials. In Chapter XI the author introduces certain generalizations of multiplicative functions and as an example considers the number of representations of an integer as sum of a given number of squares. The next chapter is devoted to Ramanujan’s \(\tau\)- function. The author uses modular forms to prove its multiplicativity and mentions the congruences satisfied by it. In Chapter XIII Lehmer’s specially multiplicative functions are treated and in Chapter XIV we encounter Ramanujan expansions of arithmetical functions. In Chapter XV a certain linear algebra associated with residue classes is considered and Chapter XVI is devoted to periodic functions. The last chapter sketches the theory of functions defined on the set of polynomials over a finite field.

Each chapter has many exercises and its own list of references. There is also an additional bibliography at the end of the book.

Unfortunately the book contains several inaccuracies and misprints. Theorems 6 and 11 have incomplete proofs. Theorem 40 is false (unless one assumes that r is a power of p), as well as the statement about group decomposition on the top of p. 154 (where one should add that the group in question is Abelian). The important fact that multiplicative functions form a group under the Dirichlet convolution is stated and used several times, but only a part of it is proved (Theorem 22). The same property of the unitary convolution is applied on p. 53, also without proof.

The first part (chapters I-VI) starts with introducing four principal convolution products (Dirichlet, Cauchy, Lucas and the unitary product), brings their main properties and then studies various divisor functions, Euler’s totient and the Moebius function, as well as several of their generalizations. Chapter VII introduces multiplicative functions in two variables and in Chapter VIII certain such functions associated with Abelian groups are studied. The next two chapters deal with Ramanujan sums and cyclotomic polynomials. In Chapter XI the author introduces certain generalizations of multiplicative functions and as an example considers the number of representations of an integer as sum of a given number of squares. The next chapter is devoted to Ramanujan’s \(\tau\)- function. The author uses modular forms to prove its multiplicativity and mentions the congruences satisfied by it. In Chapter XIII Lehmer’s specially multiplicative functions are treated and in Chapter XIV we encounter Ramanujan expansions of arithmetical functions. In Chapter XV a certain linear algebra associated with residue classes is considered and Chapter XVI is devoted to periodic functions. The last chapter sketches the theory of functions defined on the set of polynomials over a finite field.

Each chapter has many exercises and its own list of references. There is also an additional bibliography at the end of the book.

Unfortunately the book contains several inaccuracies and misprints. Theorems 6 and 11 have incomplete proofs. Theorem 40 is false (unless one assumes that r is a power of p), as well as the statement about group decomposition on the top of p. 154 (where one should add that the group in question is Abelian). The important fact that multiplicative functions form a group under the Dirichlet convolution is stated and used several times, but only a part of it is proved (Theorem 22). The same property of the unitary convolution is applied on p. 53, also without proof.

Reviewer: W.Narkiewicz

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11A25 | Arithmetic functions; related numbers; inversion formulas |