Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

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     

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.     

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     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

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.     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

Probabilistic Number Theory in Additive Arithmetic Semigroups, III

We extend the investigation of quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups given in our previous paper. Then the new and old quantitative mean-value theorems are applied to the investigation of local distribution of values of a special additive function *(a). The result is unexpected from the point of view of classical number theory. This reveals the fact that the essential divergence of the theory of additive arithmetic semigroups from classical number theory is not related to the existence of a zero of the zeta function Z(y) at y = –q ^–1.     

Ring structures on groups of arithmetic functions

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powersfα for α∈Bin(R) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.     

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C³ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C¹ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.     

NOTE ON BRIDGING THEOREMS FOR SEMISIMPLICIAL SETS

Abstract

In this note we present a variant of bridging theorems in abstract homotopy theory which applies to the category of semisimplicial sets yielding bridging theorems for semisimplicial sets which might prove useful in semi-algebraic topology.     

Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration

A detailed study of digital (t, m, s)-nets and digital (T,s)-sequences constructed over finite rings is carried out. We present general existence theorems for digital nets and sequences and also explicit constructions. Special attention is devoted to the case where the finite ring is a residue class ring of the integers. This study is motivated by the problem of numerical integration of multivariate Walsh series by quasi-Monte Carlo methods, for which we also provide a general error bound.The third author was supported by the CEI Project PACT, WP5.1.2.1.3.     

COHOMOLOGICAL DIMENSION OF UNIFORM SPACES

Abstract

In this paper we obtain classification and extension theorems for uniform spaces, using the ?ech cohomology theory based on the finite uniform coverings, and study the associated cohomological dimension theory. In particular, we extend results for the cohomological dimension theory on compact Hausdorff spaces or compact metric spaces to those for our cohomological dimension theory on uniform spaces.     

THEOREMS AD INFINITUM

Abstract

The theorems of a first order theory can be partially ordered according to their strength. As a Consequence of two famous theorems of Gödel. the order turns out to be dense. This consequence is either disastrous or amusing, according to your personal view of research in mathematics.     

JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY,AND THE THEORY OF ARITHMETICAL FUNCTIONS

Abstract

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

In this paper some important contributions of John Knopfmacher to ‘Abstract Analytic Number Theory’ are described. This theory investigates semigroups with countably many generators (generalized ‘primes’), with a norm map (or a ‘degree map’), and satisfying certain conditions on the number of elements with norm less than x (Axiom A resp. Axiom A#), and ‘arithmetical’ functions defined on these semigroups.

It is tried to show some of the impact of John Knopfmachers ideas to the future development of number theory, in particular for the topics ‘arithmetical functions’ and ‘asymptotics in additive arithmetical semigroups’.     

ℤ2-Graded Number Theory

Let ? be the set of pairs of integers, together with addition and multiplication as given in (1) and (2) below. The arithmetics of ? reflects a certain arithmetics of characters of symmetric groups, whose corresponding Young diagrams are supported on hooks. This arithmetics gives rise to a ?₂-graded (or super or hyperbolic) number theory. Many theorems from number theory have their ?₂-graded analogues in ?. Here we study a few basic aspects of that theory.     

Multiplicative perturbation of nonlinear m-accretive operators

Criteria are obtained for when an accretive product (i.e., composition) BA of nonlinear m-accretive operators A and B in a Banach space X will be itself m-accretive; and, in particular, when a monotone product of two maximal monotone operators in a Hilbert space will be maximal monotone. This extends the theory of multiplicative perturbation of infinitesimal generators of contraction semigroups to the nonlinear case. Also obtained as a biproduct are existence theorems for certain Hammerstein integral equations.     

Representation theorems for directed completions of consistent algebraic L-domains

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory. Received January 8, 2005; accepted in final form June 15, 2005.     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.