A modified form of Siegel's mean-value theorem

1955 ◽  
Vol 51 (4) ◽  
pp. 565-576 ◽  
Author(s):  
A. M. Macbeath ◽  
C. A. Rogers
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Möbius Function ◽  
Mean Value Theorem ◽  
Star Body ◽  
Mobius Function ◽  
Whole Space ◽  
Three Stages ◽  
Image Position

The Minkowski–Hlawka theorem† asserts that, if S is any n-dimensional star body, with the origin o as centre, and with volume less than 2ζ(n), then there is a lattice of determinant 1 which has no point other than o in S. One of the methods used to prove this theorem splits up into three stages, (a) A function ρ(x) is considered, and it is shown that some suitably defined mean value of the sumtaken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integralover the whole space. (b) By taking ρ(x) to be equal, or approximately equal, towhere σ(x) is the characteristic function of S, and μ(r) is the Möbius function, it is shown that a corresponding mean value of the sumwhere Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

scholarly journals A natural proof of the cyclotomic identity

1990 ◽  
Vol 42 (2) ◽  
pp. 185-189 ◽  
Author(s):  
D.E. Taylor
Keyword(s):  
Natural Isomorphism ◽  
Möbius Function ◽  
Mobius Function ◽  
Image Position

The cyclotomic identitywhere and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

scholarly journals An Analogue of a Result of Jacobsthal

1962 ◽  
Vol 13 (2) ◽  
pp. 139-142 ◽  
Author(s):  
Eckford Cohen
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Image Position ◽  
Positive Integers

Jacobsthal (4)has proved that the n×n matrixis invertible with the inverse,Here μ(x) denotes the Möbius function for positive integral x and is assumed to be 0 for other values; [x] has its usual meaning as the number of positive integers ≦x.

scholarly journals A local mean value theorem for the complex plane

1969 ◽  
Vol 16 (4) ◽  
pp. 329-331 ◽  
Author(s):  
J. M. Robertson
Keyword(s):  
Complex Plane ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Local Mean ◽  
Rouche’S Theorem ◽  
Image Position ◽  
Rouché's Theorem

The equationneed not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

A Congruence for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (05) ◽  
pp. 571-574 ◽  
Author(s):  
M.V. Subbarao
Keyword(s):  
Fixed Points ◽  
Möbius Function ◽  
Arithmetic Functions ◽  
Left Member ◽  
Mobius Function ◽  
Image Position ◽  
Positive Integers

There is considerable literature concerning the century old result that for arbitrary positive integers a and m, 1.1 where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.

scholarly journals On (Not) Computing the Möbius Function Using Bounded Depth Circuits

2012 ◽  
Vol 21 (6) ◽  
pp. 942-951 ◽  
Author(s):  
BEN GREEN
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Formula Group ◽  
Image Position ◽  
Bounded Depth Circuits

Any function F: {0,. . ., N − 1} → {−1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Möbius function μ in the sense that \[ \frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) &#x2192; 0 \quad\text{as}~~ N &#x2192; \infty. \] The proof combines a result of Linial, Mansour and Nisan with techniques of Kátai and Harman, used in their work on finding primes with specified digits.

A simpler proof of Levinson's theorem

1985 ◽  
Vol 97 (3) ◽  
pp. 385-395 ◽  
Author(s):  
J. B. Conrey ◽  
A. Ghosh
Keyword(s):  
Functional Equation ◽  
Critical Line ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Levinson’S Theorem ◽  
The Mean ◽  
Image Position

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], letwhere Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and letwhereandwhere

scholarly journals Some Generalizations of an Identity of Subhankulov

1977 ◽  
Vol 20 (4) ◽  
pp. 489-494
Author(s):  
D. Suryanarayana ◽  
David T. Walker
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Ramanujan’S Sum ◽  
Image Position ◽  
Ramanujan's Sum

AbstractIn 1957, M. A. Subhankulov established the following identitywhere ; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.

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