3-adic averages of counting functions

1998 ◽  
pp. 76-95
Author(s):  
Günther J. Wirsching
Keyword(s):  
Counting Functions

A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

2005 ◽  
Vol 01 (04) ◽  
pp. 563-581 ◽  
Author(s):  
A. KNOPFMACHER ◽  
M. E. MAYS
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Recursive Algorithms ◽  
Additive Number ◽  
Combinatorial Objects ◽  
Survey Techniques ◽  
Counting Functions ◽  
General Field ◽  
Positive Integers

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

Scalar Products of Certain Hecke L-Series and Moments of Weighted Norm-Counting Functions

1985 ◽  
Vol 28 (3) ◽  
pp. 272-279 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Dirichlet Series ◽  
Finite Groups ◽  
Tensor Products ◽  
Meromorphic Continuation ◽  
Weighted Norm ◽  
Mellin Transformation ◽  
Counting Functions ◽  
Representations Of Finite Groups ◽  
Scalar Products

AbstractWe consider Dirichlet series R(s), constructed by taking scalar products of Hecke L-series with ray-class characters. Using a theorem of G. W. Mackey on tensor products of representations of finite groups we show that R(s) has a meromorphic continuation into Re(s) > 1/2 (obtained by more sophisticated methods in [l]-[5]); we then obtain estimates for the growth of R(s) on vertical lines. Via the Mellin transformation we deduce asymptotics for various weighted moment sums involving ideals of given ray-class and norm, in one or several fields simultaneously.

scholarly journals Orbital L-functions for the Space of Binary Cubic Forms

2013 ◽  
Vol 65 (6) ◽  
pp. 1320-1383 ◽  
Author(s):  
Takashi Taniguchi ◽  
Frank Thorne
Keyword(s):  
Functional Equations ◽  
Companion Paper ◽  
Intrinsic Interest ◽  
Counting Functions ◽  
Cubic Fields ◽  
Cubic Forms

AbstractWe introduce the notion of orbital L-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

2000 ◽  
Vol 6 (3) ◽  
pp. 311-330 ◽  
Author(s):  
Jan Krajíček ◽  
Thomas Scanlon
Keyword(s):  
Euler Characteristic ◽  
Order Structure ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
Open Problems ◽  
Partially Ordered ◽  
Counting Functions ◽  
Finite Structures ◽  
Definable Sets ◽  
Grothendieck Rings

AbstractWe recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

scholarly journals Relations between counting functions on free groups and free monoids

2018 ◽  
Vol 12 (4) ◽  
pp. 1485-1521
Author(s):  
Tobias Hartnick ◽  
Alexey Talambutsa
Keyword(s):  
Free Groups ◽  
Free Monoids ◽  
Counting Functions
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