A short remark on an arithmetic function

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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Generating Functions for a Class of Arithmetic Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1966-051-9 ◽

1966 ◽

Vol 9 (4) ◽

pp. 427-431 ◽

Cited By ~ 3

Author(s):

A. A. Gioia ◽

M.V. Subbarao

Keyword(s):

Generating Functions ◽

Multiplicative Function ◽

Arithmetic Function ◽

Arithmetic Functions ◽

Prime Divisors ◽

Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

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Mean value theorems for a class of density-like arithmetic functions

International Journal of Number Theory ◽

10.1142/s1793042121500214 ◽

2020 ◽

pp. 1-15

Author(s):

Lucas Reis

Keyword(s):

Finite Field ◽

Arithmetic Function ◽

Field Extension ◽

Mean Value ◽

Arithmetic Functions ◽

Mean Value Theorem ◽

Primitive Elements ◽

Mean Values ◽

Mean Value Theorems ◽

Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

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RUELLE TYPE ZETA FUNCTIONS FOR TORI AND SOME ARITHMETICS

International Journal of Mathematics ◽

10.1142/s0129167x0400251x ◽

2004 ◽

Vol 15 (07) ◽

pp. 691-715 ◽

Cited By ~ 1

Author(s):

NOBUSHIGE KUROKAWA ◽

MASATO WAKAYAMA

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Distribution ◽

Arithmetic Function ◽

Zeta Functions ◽

Arithmetic Functions ◽

Length Functions ◽

Euler Products

We introduce various Ruelle type zeta functions ζL(s) according to a choice of homogeneous "length functions" for a lattice L in [Formula: see text] via Euler products. The logarithm of each ζL(s) yields naturally a certain arithmetic function. We study the asymptotic distribution of averages of such arithmetic functions. Asymptotic behavior of the zeta functions at the origin s=0 are also investigated.

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A note on the Entscheidungsproblem

Journal of Symbolic Logic ◽

10.2307/2269326 ◽

1936 ◽

Vol 1 (1) ◽

pp. 40-41 ◽

Cited By ~ 253

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)].

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On average values of arithmetic functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100017990 ◽

1941 ◽

Vol 37 (4) ◽

pp. 358-372 ◽

Cited By ~ 8

Author(s):

E. Fogels

Keyword(s):

Arithmetic Function ◽

Arithmetic Functions ◽

Average Order ◽

Image Position

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we haveandas x → ∞.

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A generalization of the practical numbers

International Journal of Number Theory ◽

10.1142/s1793042118500902 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1487-1503

Author(s):

Nicholas Schwab ◽

Lola Thompson

Keyword(s):

Positive Integer ◽

Arithmetic Function ◽

Arithmetic Functions ◽

Asymptotic Density ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

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Some Results on Uniform Arithmetic Circuit Complexity

DAIMI Report Series ◽

10.7146/dpb.v20i343.7969 ◽

1991 ◽

Vol 20 (343) ◽

Author(s):

Gudmund Skovbjerg Frandsen ◽

Mark Valence ◽

David Mix Barrington

Keyword(s):

Arithmetic Function ◽

Circuit Complexity ◽

Arithmetic Circuits ◽

Arithmetic Functions ◽

Constant Depth ◽

Arithmetic Circuit ◽

Arithmetic Complexity ◽

Polynomial Size ◽

Threshold Circuits ◽

Class Of Functions

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F_(2)n may be identified with an <em>n</em>-input and an n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

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On a subgroup of the group of multiplicative arithmetic functions

Journal of the Australian Mathematical Society ◽

10.1017/s144678870002070x ◽

1975 ◽

Vol 20 (3) ◽

pp. 348-358 ◽

Cited By ~ 6

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.

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On the Equilibrium Problem of a Soft Network Shell in the Presence of Several Point Loads

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.392.188 ◽

2013 ◽

Vol 392 ◽

pp. 188-190 ◽

Cited By ~ 22

Author(s):

Ildar B. Badriev ◽

Victor V. Banderov ◽

O.A. Zadvornov

Keyword(s):

Existence Theorem ◽

Equilibrium Problem ◽

Explicit Form ◽

Integral Identity ◽

Auxiliary Problem ◽

Delta Function ◽

Spatial Equilibrium ◽

External Point ◽

The Right ◽

Generalized Statement

We consider a spatial equilibrium problem of a soft network shell in the presence of several external point loads concentrated at some pairwise distinct points. A generalized statement of the problem is formulated in the form of integral identity. Then we introduce an auxiliary problem with the right-hand side given by the delta function. For the auxiliary problem we are able to find the solution in an explicit form. Due to this, the generalized statement of the problem under consideration is reduced to finding the solution of the operator equation. We establish the properties of the operator of this equation (boundedness, continuity, monotonicity, and coercitivity), which makes it possible to apply known general results from the theory of monotone operatorsfor the proof of the existence theorem. It is proved that the set of solutions of the generalized problem is non-empty, convex, and closed.

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