##
**Handbook of number theory II.**
*(English)*
Zbl 1079.11001

Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-2546-7/hbk). 637 p. (2004).

Similarly to the Handbook of Number Theoryby D. Mitrinović, J. Sándor and B. Crstici (Kluwer) (1995; Zbl 0862.11001), it is the aim of this monograph to give, in an encyclopedic manner, surveys on some topics of Number Theory with many hints to the literature.

From the Preface: “The aim of this book is to systematize and survey in an easily accessible manner the most important results from some parts of Number Theory …. Each chapter can be viewed as an encyclopedia of the considered field. …This book focuses too, as the former volume, on some important arithmetic functions …, such as Euler’s totient \(\varphi(n) \), …, \(\sigma(n) \) with the many old and new issues on Perfect Numbers; the Möbius function ……. The last chapter shows perhaps most strikingly the cross-fertilization of Number Theory with Combinatorics, Numerical mathematics, or Probability theory.”

So, this monograph deals with topics from number theory, which we are going to try to describe now.

Perfect Numbers (some history; even perfect numbers; many results for odd perfect numbers). The chapter ends with generalizations of the notion of perfect number [multiply perfect, quasi-perfect, almost perfect, super-perfect, pseudo-perfect, unitary perfect, …] and of amicable numbers. There are more than 300 references.

Generalizations and extensions of the Möbius function. This chapter is of a rather elementary nature. Many generalizations of the Möbius function (and of the notion of convolution) are given, including the Möbius function of arithmetical semigroups. There are more than 200 references.

The many facets of Euler’s totient. There are connections with prime number theory (proof of the infinitude of primes by E. E. Kummer, the exact formula \(\pi(n) = \sum_{2\leq k \leq n} [ \frac{\varphi(n)}{k-1}]\) of Vassil’ev, and other such formulae), many identities (by Liouville, Cesáro, and others), Redmond’s result \[ \prod_{n=1}^\infty \biggl( \frac{L_n}{\sqrt{5\,}} \cdot F_n\biggr) {\frac{\varphi(n)}n} = e^{-\frac{1+\sqrt5}{2\sqrt5}}. \] Next, the author deals with enumeration problems, with Fourier coefficients of even functions (E. Cohen), with algebraic independence of certain arithmetical functions, with congruence properties (the Fermat-Euler theorem \(a^{\varphi(m)} \equiv 1 \bmod m\), if \(\gcd(a,m)=1\), and many later results), Carmichael’s function (minimal order, average order), iterates of \(\varphi\), the behaviour in residue classes, for example \[ \#\{ n\leq x, \;\varphi(n) \equiv r \bmod m\} \sim c_r \cdot \frac x{\sqrt{\log x}} \] for \(m=12, r = 4, 8\) (and further results), prime totatives, the function \( n\mapsto n - \varphi(n) \), Euler’s functions \(\min\{ k\geq 1, \;n \mid \varphi(k) \}\) and \(\max \{ k\geq 1, \;\varphi(k) \mid n\}\). Next there are results on equations of the type \(\varphi(x+k) = \varphi(x) \), on the the equation \(\varphi(x)=k\), on the solvability of equations involving \(\varphi\) and other arithmetical functions (for example the result of Erdős, that the system \(\varphi(x)=\varphi(y), \; d(x) = d(y), \; \sigma(x)=\sigma(y)\) has infinitely many [non-trivial] solutions), results on the composition of \(\varphi\) with other functions, the distribution of totatives, …. Next, there are many results on cyclotomic polynomials (irreducibility, divisibility properties, the coefficients of these), and on matrices and determinants connected with \(\varphi\). Finally, many generalizations of \(\varphi\) are dealt with. There is a most valuable list of nearly 500 references, from Gauss (1801), Dedekind (1857) up to the most recent results.

Special Arithmetic Functions. “The aim of this chapter is to study some other functions which …are not so well-known, and are scattered in various fields of study ….”

Examples are the maximum (and minimum) exponent \(H(n)= \max\{ a_1, \dots, a_r\}\), where \(n = \prod_\rho p_\rho^{a_\rho^{(n)}}\), the product of exponents \(\beta(n) = a_1 \cdots a_r\), the function \(n \mapsto a_\rho(n) \), the study of consecutive prime divisors of a number, the consecutive divisors of a number, Hooley’s function \[ \Delta(n) = \max_{x\in {\mathbb R}} \#\{ d \mid n,\;x < \log d \leq x+1\}, \] and divisors in residue classes. There is a long subsection on arithmetic functions associated to the digits of a number, including \(q\)-additive and \(q\)-multiplicative functions. 360 references are given.

Stirling, Bell, Bernoulli, Euler and Eulerian Numbers. To describe the contents of this chapter in short, we quote from the introduction: “This Chapter is divided into two major parts: Stirling and Bell numbers, and Bernoulli, Euler and Eulerian numbers. These classical topics occur in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, number theory, and probability theory. Our aim is to study the many aspects of these numbers, and to point out important connections or applications in number theory and related fields. …” There are again nearly 500 references to the literature.

The reviewer thinks that this monograph is a great help to number theorists working in the fields mentioned above.

From the Preface: “The aim of this book is to systematize and survey in an easily accessible manner the most important results from some parts of Number Theory …. Each chapter can be viewed as an encyclopedia of the considered field. …This book focuses too, as the former volume, on some important arithmetic functions …, such as Euler’s totient \(\varphi(n) \), …, \(\sigma(n) \) with the many old and new issues on Perfect Numbers; the Möbius function ……. The last chapter shows perhaps most strikingly the cross-fertilization of Number Theory with Combinatorics, Numerical mathematics, or Probability theory.”

So, this monograph deals with topics from number theory, which we are going to try to describe now.

Perfect Numbers (some history; even perfect numbers; many results for odd perfect numbers). The chapter ends with generalizations of the notion of perfect number [multiply perfect, quasi-perfect, almost perfect, super-perfect, pseudo-perfect, unitary perfect, …] and of amicable numbers. There are more than 300 references.

Generalizations and extensions of the Möbius function. This chapter is of a rather elementary nature. Many generalizations of the Möbius function (and of the notion of convolution) are given, including the Möbius function of arithmetical semigroups. There are more than 200 references.

The many facets of Euler’s totient. There are connections with prime number theory (proof of the infinitude of primes by E. E. Kummer, the exact formula \(\pi(n) = \sum_{2\leq k \leq n} [ \frac{\varphi(n)}{k-1}]\) of Vassil’ev, and other such formulae), many identities (by Liouville, Cesáro, and others), Redmond’s result \[ \prod_{n=1}^\infty \biggl( \frac{L_n}{\sqrt{5\,}} \cdot F_n\biggr) {\frac{\varphi(n)}n} = e^{-\frac{1+\sqrt5}{2\sqrt5}}. \] Next, the author deals with enumeration problems, with Fourier coefficients of even functions (E. Cohen), with algebraic independence of certain arithmetical functions, with congruence properties (the Fermat-Euler theorem \(a^{\varphi(m)} \equiv 1 \bmod m\), if \(\gcd(a,m)=1\), and many later results), Carmichael’s function (minimal order, average order), iterates of \(\varphi\), the behaviour in residue classes, for example \[ \#\{ n\leq x, \;\varphi(n) \equiv r \bmod m\} \sim c_r \cdot \frac x{\sqrt{\log x}} \] for \(m=12, r = 4, 8\) (and further results), prime totatives, the function \( n\mapsto n - \varphi(n) \), Euler’s functions \(\min\{ k\geq 1, \;n \mid \varphi(k) \}\) and \(\max \{ k\geq 1, \;\varphi(k) \mid n\}\). Next there are results on equations of the type \(\varphi(x+k) = \varphi(x) \), on the the equation \(\varphi(x)=k\), on the solvability of equations involving \(\varphi\) and other arithmetical functions (for example the result of Erdős, that the system \(\varphi(x)=\varphi(y), \; d(x) = d(y), \; \sigma(x)=\sigma(y)\) has infinitely many [non-trivial] solutions), results on the composition of \(\varphi\) with other functions, the distribution of totatives, …. Next, there are many results on cyclotomic polynomials (irreducibility, divisibility properties, the coefficients of these), and on matrices and determinants connected with \(\varphi\). Finally, many generalizations of \(\varphi\) are dealt with. There is a most valuable list of nearly 500 references, from Gauss (1801), Dedekind (1857) up to the most recent results.

Special Arithmetic Functions. “The aim of this chapter is to study some other functions which …are not so well-known, and are scattered in various fields of study ….”

Examples are the maximum (and minimum) exponent \(H(n)= \max\{ a_1, \dots, a_r\}\), where \(n = \prod_\rho p_\rho^{a_\rho^{(n)}}\), the product of exponents \(\beta(n) = a_1 \cdots a_r\), the function \(n \mapsto a_\rho(n) \), the study of consecutive prime divisors of a number, the consecutive divisors of a number, Hooley’s function \[ \Delta(n) = \max_{x\in {\mathbb R}} \#\{ d \mid n,\;x < \log d \leq x+1\}, \] and divisors in residue classes. There is a long subsection on arithmetic functions associated to the digits of a number, including \(q\)-additive and \(q\)-multiplicative functions. 360 references are given.

Stirling, Bell, Bernoulli, Euler and Eulerian Numbers. To describe the contents of this chapter in short, we quote from the introduction: “This Chapter is divided into two major parts: Stirling and Bell numbers, and Bernoulli, Euler and Eulerian numbers. These classical topics occur in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, number theory, and probability theory. Our aim is to study the many aspects of these numbers, and to point out important connections or applications in number theory and related fields. …” There are again nearly 500 references to the literature.

The reviewer thinks that this monograph is a great help to number theorists working in the fields mentioned above.

Reviewer: Wolfgang Schwarz (Frankfurt / Main)

### MSC:

11-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to number theory |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

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

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

11B68 | Bernoulli and Euler numbers and polynomials |

11B73 | Bell and Stirling numbers |

11N37 | Asymptotic results on arithmetic functions |

11N64 | Other results on the distribution of values or the characterization of arithmetic functions |

### Keywords:

arithmetic functions; Euler’s \(\varphi\)–function; Möbiusfunction; perfect numbers; convolutions; Carmichael’s function; composition of arithmetical functions; Stirling Numbers; Bell Numbers; consecutive divisors; special arithmetical functions### Citations:

Zbl 0862.11001### Online Encyclopedia of Integer Sequences:

Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.

Product of divisors of n.

Product of the proper divisors of n.

Binary self or Colombian numbers: numbers that cannot be expressed as the sum of distinct terms of the form 2^k+1 (k>=0), or equivalently, numbers not of form m + sum of binary digits of m.

Base 4 self or Colombian numbers (not of form k + sum of base 4 digits of k).

Base 6 self or Colombian numbers (not of form k + sum of base 6 digits of k).

Base 8 self or Colombian numbers (not of form k + sum of base 8 digits of k).

Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

a(n) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.

Number of positive integers, k, where k <= n and gcd(k,n) = gcd(k+1,n) = 1.

Solutions k of the equation phi(k) = phi(k-1) + phi(k-2). Also known as Phibonacci numbers.

Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.

Numbers n such that n = phi(x)*core(x) for some x <= n, where phi(x) is the Euler totient function and core(x) the squarefree part of x.

Perfect totient numbers.

Decimal expansion of average deviation of the total number of prime factors.

Decimal expansion of Silverman’s constant.

2-Suzanne numbers.

3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

2-Monica numbers.

a(n) is the number of positive integers k which are <= n and where k, k-1 and k+1 are each coprime to n.

3-Monica numbers.

Numbers that are of the form k + sum of binary digits of k for some nonnegative integer k.

a(n) = n + (sum of digits in base-3 representation of n).

a(n) = n + (sum of digits in base-5 representation of n).

Decimal expansion of the binary self-numbers density constant.

Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

Numbers k such that Sum_{d|k} nphi(d) = k where the sum is over nonunitary divisors of k and nphi(k) is the nonunitary totient function (A254503).

Exponential superperfect numbers (or e-superperfect numbers): numbers m such that esigma(esigma(m)) = 2m, where esigma(m) is the sum of exponential divisors of m (A051377).

Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).

Odd numbers k such that s(k) = s(k+2), where s(k) is Schemmel’s totient function of order 2 (A058026).

Numbers k such that both k and k + 1 are Niven numbers in base 2 (A049445).

Starts of runs of 3 consecutive Niven numbers in base 2 (A049445).

Starts of runs of 4 consecutive Niven numbers in base 2 (A049445).

Numbers divisible by the minimal exponent in their prime factorization (A051904).

Numbers divisible by the maximal exponent in their prime factorization (A051903).

Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).

Numbers m such that k + A005361(k) <= m for all k < m.

Zeckendorf self numbers: numbers not of the form k + A007895(k).

Dual-Zeckendorf self numbers: numbers not of the form k + A112310(k).

Phi-base self numbers: positive numbers not of the form k + A055778(k).

Factorial-base self numbers: numbers not of the form k + A034968(k).

Primorial-base self numbers: numbers not of the form k + A276150(k).

Number k > 1 such that gpf(phi(k)/lambda(k)) = A006530(A000010(k)/A002322(k)) > log(log(k))^2.

a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

Self numbers in base i-1: numbers not of the form k + A066323(k).

a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.

a(n) is the least number larger than 1 which is a self number in all the even bases b = 2*k for 1 <= k <= n.

Numbers k such that k | A002619(k).

Decimal expansion of sinh(Pi/2)/(Pi/2)^2.