It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the
Since the invertibility of the Redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express
is not invertible precisely when the Mertens function is zero (or is close to changing signs).
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 [2] where
denotes the infinite q-Pochhammer symbol and where the lower triangular matrix sequence is exactly generated as the coefficients of
, 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 to be expressed as a sum over the convolved coefficients
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).
, of these matrices is defined here without the full scope of the somewhat technical proofs justifying the bounds from the references cited above.
is Euler's classical gamma constant, and where the remaining coefficients of these polynomials are bounded by A plot of the much more size-constrained nature of the eigenvalues of
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 a great deal more combinatorial in construction than typical treatments of the special Redheffer matrix determinants.
Nonetheless, this combinatorial twist on enumerating special sequences of sums has been explored more recently in a number of papers and is a topic of active interest in pre-print archives.
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.
The inverse matrix terms are referred to a generalized Mobius function within the context of sums of this type in.
[9] First, given any two non-identically-zero arithmetic functions f and g, we can provide explicit matrix representations which encode their Dirichlet convolution in rows indexed by natural numbers
where this upper bound is the prime omega function which counts the number of distinct prime factors of n. As this example shows, we can formulate an alternate way to construct the Dirichlet inverse function values via matrix inversion with our variant Redheffer matrices,
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 whose zero-or-one valued entries denote inclusion in an increasing sequence of sets of the natural numbers
To illustrate that the previous mouthful of jargon makes good sense in setting up a matrix based system for representing a wide range of special summations, consider the following construction: Let
) 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 (and their logarithms).
The referenced article by Mousavi and Schmidt (2017) develops a factorization-theorem-like treatment to expanding these sums which is an analog to the Lambert series factorization results given in the previous section above.
These inverse matrices have many curious properties (and a good reference pulling together a summary of all of them is currently lacking) which are best intimated and conveyed to new readers by inspection.
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.