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 \[ \frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) → 0 \quad\text{as}~~ N → \infty. \] 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.

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.

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.

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 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.

The Distribution of Irreducible Polynomials in Several Indeterminates II

1965 ◽  
Vol 17 ◽  
pp. 261-266 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Möbius Function ◽  
Irreducible Polynomials ◽  
Mobius Function ◽  
Image Position

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by(1.1)where μ(r) is the Möbius function. It follows from (1) that(12)

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

Relations between arithmetical functions

1967 ◽  
Vol 63 (4) ◽  
pp. 1027-1029
Author(s):  
C. J. A. Evelyn
Keyword(s):  
Positive Integer ◽  
Möbius Function ◽  
Natural Numbers ◽  
Arithmetical Functions ◽  
Mobius Function ◽  
Image Position ◽  
Fixed Positive Integer

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and ifthenwhere T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that ifwherethenThus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

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