A Method of Tabulating the Number-Theoretic Function g(k)

1992 ◽  
Vol 59 (199) ◽  
pp. 251 ◽  
Author(s):  
Renate Scheidler ◽  
Hugh C. Williams
Keyword(s):  
Theoretic Function

scholarly journals The Series Σ∞1 f(n)/n for Periodic f

1965 ◽  
Vol 8 (4) ◽  
pp. 413-432 ◽  
Author(s):  
Arthur E. Livingston
Keyword(s):  
Positive Integer ◽  
Complex Valued ◽  
Theoretic Function

We are here concerned with the problem of deciding when Σ∞n=1 f(n)/n ≠ 0, given that f is periodic and the series convergent. In particular, we considerConjecture A. Let p be a positive integer and f a (real-or complex-valued) number-theoretic function with period p.

scholarly journals Iterates of a number-theoretic function

1969 ◽  
Vol 23 (105) ◽  
pp. 181-181 ◽  
Author(s):  
Mohan Lal
Keyword(s):  
Theoretic Function

scholarly journals A method of tabulating the number-theoretic function $g(k)$

1992 ◽  
Vol 59 (199) ◽  
pp. 251-251
Author(s):  
Renate Scheidler ◽  
Hugh C. Williams
Keyword(s):  
Theoretic Function

Can there be no nonrecursive functions?

1971 ◽  
Vol 36 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Joan Rand Moschovakis
Keyword(s):  
Free Choice ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Choice Sequence ◽  
Church's Thesis ◽  
Theoretic Function ◽  
Ordinary Number ◽  
Definable Function

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).

Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy

1972 ◽  
Vol 37 (2) ◽  
pp. 281-292 ◽  
Author(s):  
S. S. Wainer
Keyword(s):  
Natural Manner ◽  
Recursive Functions ◽  
Class A ◽  
Primitive Recursive ◽  
Theoretic Function ◽  
Number Class

It is well known that iteration of any number-theoretic function f, which grows at least exponentially, produces a new function f′ such that f is elementary-recursive in f′ (in the Csillag-Kalmar sense), but not conversely (since f′ majorizes every function elementary-recursive in f). This device was first used by Grzegorczyk [3] in the construction of a properly expanding hierarchy {ℰn: n = 0, 1, 2, …} which provided a classification of the primitive recursive functions. More recently it was shown in [7] how, by iterating at successor stages and diagonalizing over fundamental sequences at limit stages, the Grzegorczyk hierarchy can be extended through Cantor's second number-class. A problem which immediately arises is that of classifying all recursive functions, and an answer to this problem is to be found in the general results of Feferman [1]. These results show that although hierarchies of various types (including the above extensions of Grzegorczyk's hierarchy) can be produced, which range over initial segments of the constructive ordinals and which do provide complete classifications of the recursive functions, these cannot be regarded as classifications “from below”, since the method of assigning fundamental sequences at limit stages must be highly noneffective. We therefore adopt the more modest aim here (as in [7], [12], [14]) of characterising certain well-known (effectively generated) subclasses of the recursive functions, by means of hierarchies generated in a natural manner, “from below”.

Sign in / Sign up

Export Citation Format

Share Document