A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (1) ◽  
pp. 40-41 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Present Note ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Symbolic Logic ◽  
Recursive Definition ◽  
System L ◽  
Preceding Paragraph ◽  
Individual Variables ◽  
Definition Of ◽  
Positive Integers

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

Completeness in the theory of types

1950 ◽  
Vol 15 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Formal System ◽  
Axiom System ◽  
Second Order ◽  
True Proposition ◽  
Arbitrary Domain ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual ◽  
Positive Integers

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

Additive arithmetic functions on intervals

1988 ◽  
Vol 103 (1) ◽  
pp. 163-179 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Arithmetic Function ◽  
Value Distribution ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Positive Integers

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.

scholarly journals A study on some generalized multiplicative and generalized additive arithmetic functions

2021 ◽  
Vol 27 (1) ◽  
pp. 32-44
Author(s):  
D. Bhattacharjee ◽  
Keyword(s):  
Additive Function ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Positive Integers

In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.

scholarly journals Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

2016 ◽  
Vol 14 (1) ◽  
pp. 146-155 ◽  
Author(s):  
Siao Hong ◽  
Shuangnian Hu ◽  
Shaofang Hong
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Sets ◽  
Positive Integers

AbstractLet f be an arithmetic function and S= {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and S1, …, Sk being gcd-closed sets such that (lcm(Si), lcm(Sj)) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

scholarly journals Products of several commutators in a Lie nilpotent associative algebra

2017 ◽  
Vol 27 (08) ◽  
pp. 1027-1040 ◽  
Author(s):  
Galina Deryabina ◽  
Alexei Krasilnikov
Keyword(s):  
General Formula ◽  
Associative Algebra ◽  
Present Note ◽  
Commutator Formula ◽  
Positive Integers

Let [Formula: see text] be a field of characteristic [Formula: see text] and let [Formula: see text] be a unital associative [Formula: see text]-algebra. Define a left-normed commutator [Formula: see text] [Formula: see text] recursively by [Formula: see text], [Formula: see text] [Formula: see text]. For [Formula: see text], let [Formula: see text] be the two-sided ideal in [Formula: see text] generated by all commutators [Formula: see text] ([Formula: see text]. Define [Formula: see text]. Let [Formula: see text] be integers such that [Formula: see text], [Formula: see text]. Let [Formula: see text] be positive integers such that [Formula: see text] of them are odd and [Formula: see text] of them are even. Let [Formula: see text]. The aim of the present note is to show that, for any positive integers [Formula: see text], in general, [Formula: see text]. It is known that if [Formula: see text] (that is, if at least one of [Formula: see text] is even), then [Formula: see text] for each [Formula: see text] so our result cannot be improved if [Formula: see text]. Let [Formula: see text]. Recently, Dangovski has proved that if [Formula: see text] are any positive integers then, in general, [Formula: see text]. Since [Formula: see text], Dangovski’s result is stronger than ours if [Formula: see text] and is weaker than ours if [Formula: see text]; if [Formula: see text], then [Formula: see text] so both results coincide. It is known that if [Formula: see text] (that is, if all [Formula: see text] are odd) then, for each [Formula: see text], [Formula: see text] so in this case Dangovski’s result cannot be improved.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

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