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Wikipedia / Redheffer matrix

In mathematics, a Redheffer matrix as studied by , is a square (0,1) matrix whose entries aij are 1 if i divides j or if j=1; otherwise, aij=0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose Redheffer matrix.

Variants and definitions of component matrices

Since the invertibility is defined to be the (0,1) matrix, which plainly can be seen by an application of Mobius inversion. We then have a characterization of the singularity

If we define the function

Redheffer (transpose) matrix to be the nxn in usual matrix notation. We will continue to make use this notation throughout the next sections.

Examples

The matrix below is the 12 are marked in blue.

A corresponding application of the Mobius inversion formula Redheffer transpose matrix is always invertible, with inverse entries given by

denotes the Moebius function inverse Redheffer transpose matrix is given by

Key properties

Singularity and relations to the Mertens function and special series

Determinants

The determinant of the n × n square Redheffer matrix is given by the Mertens function M(n is not invertible precisely when the Mertens function is zero (or is close to changing signs). As a corollary of the disproof of the Mertens conjecture is singular at infinitely many natural numbers.

The determinants of the Redheffer matrices are immediately tied to the Riemann hypothesis.

Factorizations of sums encoded by these matrices

In a somewhat unconventional construction which reinterprets the (0,1) matrix entries to denote inclusion in some increasing sequence of indexing sets, we can see that these matrices are also related to factorizations of Lambert series. This observation is offered in so much as for a fixed arithmetic function f, the coefficients of the next Lambert series expansion over f provide a so-called inclusion mask for the indices over which we sum f to arrive at the series coefficients of these expansions. Notably, observe that

Now in the special case of these divisor sums, which we can see from the above expansion, are codified by Boolean (zero-one) valued inclusion in the sets of divisors of a natural number n, it is possible to re-interpret the Lambert series generating functions which enumerate these sums via yet another matrix-based construction. Namely, Merca and Schmidt (2017-2018) proved invertible matrix factorizations expanding these generating functions in the form of

denotes the infinite q-Pochhammer symbol, through these terms also have interpretations as differences of special even (odd) indexed partition functions. Merca and Schmidt (2017) also proved a simple inversion formula which allows the implicit function f of the original Lambert series generating function in the form of

where p(n) denotes the partition function is the Moebius function inherit a quadratic dependence on j through the pentagonal number theorem for the sake of completion here.

Other than that the underlying so-termed mask matrix which specifies the inclusion of indices in the divisor sums at hand are invertible, utilizing this type of construction to expand other Redheffer-like matrices for other special number theoretic sums need not be limited to those forms classically studied here. For example, in 2018 Mousavi and Schmidt extend such matrix based factorization lemmas to the cases of Anderson-Apostol divisor sums (of which Ramanujan sums are a notable special case) and sums indexed over the integers that are relatively prime to each n (for example, as classically defines the tally denoted by the Euler phi function). More to the point, the examples considered in the applications section below suggest a study of the properties of what can be considered generalized Redheffer matrices representing other special number theoretic sums.

Spectral radius and eigenspaces

when n.

Characterizing eigenvectors

is an eigenvector corresponding to some eigenvalue the following two conditions hold:

If we restrict ourselves to the so-called non-trivial we can recursively compute the remaining n-1 components according to the formula

we can define the sequences of

if and only if

Secondly, we have an established formula for the Dirichlet series, or Dirichlet generating function given by

of course as usual denotes the Riemann zeta function.

Bounds and properties of non-trivial eigenvalues

A graph theoretic interpretation to evaluating the zeros of the characteristic polynomial and bounding its coefficients is given in Section 5.1 of. Estimates of the sizes of the Jordan blocks corresponding to the eigenvalue one are given in. A brief overview of the properties of a modified and define a sequence of auxiliary polynomial expansions according to the formula

, which satisfy

is Euler's classical gamma constant, and where the remaining coefficients of these polynomials are bounded by

which are not characterized by these two dominant zeros of the polynomial seems to be remarkable as evidenced by the only 20 is available here for reference.

Applications and generalizations

We provide a few examples of the utility of the Redheffer matrices interpreted as a (0,1) matrix whose parity corresponds to inclusion in an increasing sequence of index sets. These examples should serve to freshen up some of the at times dated historical perspective of these matrices, and their being footnote-worthy by virtue of an inherent, and deep, relation of their determinants to the Mertens function and equivalent statements of the Riemann Hypothesis. This interpretation is defined above, observe that this type of expansion is in many ways essentially just another variation of the usage of a Toeplitz matrix to represent truncated power series expressions where the matrix entries are coefficients of the formal variable in the series. Let's explore an application of this particular view of a (0,1) matrix as masking inclusion of summation indices in a finite sum over some fixed function. See the citations to the references and for existing generalizations of the Redheffer matrices in the context of general arithmetic function cases. The inverse matrix terms are referred to a generalized Mobius function within the context of sums of this type in.

Matrix products expanding Dirichlet convolutions and Dirichlet inverses

First, given any two non-identically-zero arithmetic functions f and g, we can provide explicit matrix representations which encode their Dirichlet convolution:

gives the convolved Dirichlet sums

is arbitrary.

One task that is particularly onerous given an arbitrary function f is to determine its Dirichlet inverse exactly without resorting to a standard recursive definition of this function via yet another convolved divisor sum involving the same function f with its under-specified inverse to be determined:

It is clear that in general the Dirichlet inverse for f where this upper bound is the prime omega function which counts the number of distinct prime factors of n.

Generalizations of the Redheffer matrix forms: GCD sums and other matrices whose entries denote inclusion in special sets

There are several often cited articles from worthy journals that fight to establish expansions of number theoretic divisor sums, convolutions, and Dirichlet series (to name a few) through matrix representations. Besides non-trivial estimates on the corresponding spectrum and eigenspaces associated with truly notable and important applications of these representations—the underlying machinery in representing sums of these forms by matrix products is to effectively define a so-termed masking matrix be a sequence of index sets, and for any fixed arithmetic function define the sums

One of the classes of sums considered by Mousavi and Schmidt (2017) defines the relatively prime divisor sums by setting the index sets in the last definition to be

This class of sums can be used to express important special arithmetic functions of number theoretic interest, including Euler's phi function) as

and even the Mobius function through its representation as a discrete (finite) Fourier transform:

Citations in the full paper provide other examples of this class of sums including applications to cyclotomic polynomials then allow us to perform the analog of Moebius inversion for divisor sums which can be used to express the summand functions f and the relevant matrices defined for this case given as follows:

Examples of invertible matrices which define other special sums with non-standard, however, clear applications should be catalogued and listed in this generalizations section for completeness. An existing summary of inversion relations, and in particular, exact criteria under which sums of these forms can be inverted and related is found in many references on orthogonal polynomials. Other good examples of this type of factorization treatment to inverting relations between sums over sufficiently invertible, or well enough behaved triangular sets of weight coefficients include the Mobius inversion formula, the binomial transform, and the Stirling transform, among others.

See also

References