CONTRIBUTIONS TO ABSTRACT ANALYTIC NUMBER THEORY

Abstract

This paper reports on some recent contributions to the theory of multiplicative arithmetic semigroups, which have been initiated by John Knopfmacher's work on abstract analytic number theory. They concern weighted inversion theorems of the Wiener type, mean-value theorems for multiplicative functions, and Ramanu-jan expansions.     

Generalizations of Alladi’s formula for arithmetical semigroups

In this article, we prove that a general version of Alladi’s formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom \(A^{\#}\). As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry, and graph theory.

     

KNOPFMACHER EXPANSIONS IN NUMBER THEORY

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We survey in this paper J. Knopfmacher results on number theoretical expansions as they arised from our common collaboration.     

Arithmetical semigroups III: Elements with prime factors in residue classes

We study the distribution of elements in an additive arithmetical semigroup (G, ) (as introduced by John Knopfmacher) in whose canonical decomposition the degrees of the prime elements belong to a given union of residue classes modk.     

q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial r?ole.     

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

 Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ? _q [X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ? (Received 25 August 1998; in revised form 7 June 1999)     

DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS DEFINED ON SEMIGROUPS

Abstract

The value distribution problem for real-valued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of number-theoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions only under some extra condition. In this way, applying the known results for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.     

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

 Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ? _q [X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ?     

Arithmetical semigroups V: Multiplicative functions

Several classical results on multiplicative functions ℕ → ℂ are transposed to multiplicative functionsG → ℂ where (G, σ) denotes an additive arithmetical semigroup as introduced by John Knopfmacher.     

Mean-Value Theorems in Arithmetic Semigroups

This paper reports on recent progress in the theory of multiplicative arithmetic semigroups, which has been initiated by John Knopfmacher's work on abstract analytic number theory. In particular, it deals with abstract versions of the mean-value theorems of Delange, of Wirsing, and of Halász for multiplicative functions on arithmetic semigroups G with Axiom A . The Turán Kubilius inequality is transferred to G , and methods developed by Rényi, Daboussi and Indlekofer, Lucht and Reifenrath are utilized. As byproduct a new proof of the abstract prime number theorem is obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory. Received October 18, 2001; in final form April 11, 2002     

SOME CONSEQUENCES OF THE TARSKI-KANTOROVITCH ORDERING THEOREM IN METRIC FIXED POINT THEORY

Abstract

We show that the Tarski-Kantorovitch Principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.     

Arithmetical semigroups I: direct factors

There are two main theorems due to P. Erdös et al. on direct factors of IN. They are concerned with the asymptotic density and the distribution of primes. The concept of a direct factor is carried over to arithmetical semigroups as they were introduced by J. Knopfmacher and the two corresponding main theorems are stated and proved.     

On Sums of Roots of Unity

 Suppose that for some root of unity ζ of order Q with and all coefficients a _i belonging to a number field L. We bound Q in terms of k and . This generalizes a result of Conway and Jones for the case of rational coefficients. Moreover, we give an application to linear relations among characteristic functions of arithmetical progressions. (Received 18 January 1999; in revised from 14 June 1999)     

On the number of large oscillations of some arithmetical power series

We apply a method of J. Kaczorowski in order to study the number of large oscillations of the error term of weighted averages of some arithmetical functions. Received: 16 April 2002     

On the Distribution of Coefficients of Logarithmic Derivativesof L-Functions Attached to Certain Arithmetical Semigroups

 Under the Riemann Hypothesis for the classical Riemann zeta function, there exist infinitely many arithmetically non-isomorphic arithmetical semigroups with the property that one of the associated L-functions vanishes at . Moreover, there are no restrictions in the distribution of prime divisors of a given norm except an obvious one concerning the order of magnitude. Received 22 December 1997 in revised form 12 May 1998     

Summatory Formula of the Convolution of Two Arithmetical Functions

 We study the asymptotic formula of for some arithmetical functions f and g. This generalizes the case investigated by Balakrishnan and Pétermann. Received 15 January 2001; in revised form 7 July 2001     

ON PRIME RINGS AND THE RADICALS ASSOCIATED WITH THEIR DEGREES OF PRIMENESS

Abstract

Prime ringsMaybe classified by the sizes of the sets that ‘insulate’ their elements from annihilation. For a cardinal m > 0, the class [Pbar]_r,(m) of all rings that are right prime of ‘bound at most m’ is studied, with particular reference to its closure under constructions such as matrix rings, semigoup rings, orders and extensions. The classes [Pbar]_r,(m) are special in the sense of radical theory for each m > 0. The attendant upper radicals υ[Pbar]_r,(m) are right (and not left) strong; their compatibility with certain ring constructions is examined. In the lattice of radicals (where they form a strictly descending chain), their positions are described, relative to various familiar radicals.     

Probabilistic number theory in additive arithmetic semigroups II

We prove two quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups. On the basis of the two theorems, a central limit theorem of additive functions on additive arithmetic semigroups is proved with a best possible error estimate. This generalizes the vital results of Halász and Elliott in classical probabilistic number theory to function fields. Received October 26, 1998; in final form April 5, 2000 / Published online October 11, 2000