A Redheffer matrix is a (0,1)-matrix that is square and has elements
equal to 1 if
or
(divides), and 0 otherwise. It is an integer
matrix and hence a rational matrix. For , 2, ..., the first few Redheffer matrices
are
The Redheffer matrix of order 255 is illustrated above.
The determinant of the Redheffer matrix is equal to the Mertens
function.
For ,
2, ..., the first few values are therefore 1, 0, , , , , , , , ... (OEIS A002321).
for
(Vaughan 1993, p. 286; Vaughan 1996; Trott 2004, p. 57), where denotes the binary logarithm
and
is the floor function. For , . The first few values for , 2, ... are 1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11,
... (OEIS A083058).
Sloane, N. J. A. Sequences A002321/M0102 and A083058 in "The On-Line Encyclopedia
of Integer Sequences."Trott, M. The
Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.Vaughan,
R. C. "On the Eigenvalues of Redheffer's Matrix. I." In Number
Theory with an Emphasis on the Markoff Spectrum (Ed. A. D. Pollington
and W. Moran). New York: Dekker, pp. 283-296, 1993.Vaughan,
R. C. "On the Eigenvalues of Redheffer's Matrix. II." J. Austral.
Math. Soc.60, 260-273, 1996. https://doi.org/10.1017/S1446788700037654.
equal to 1 if 
or 
(
divides 
), and 0 otherwise. It is an integer
matrix and hence a rational matrix. For 
, 2, ..., the first few Redheffer matrices
are
![[1],[1 1; 1 1],[1 1 1; 1 1 0; 1 0 1],[1 1 1 1; 1 1 0 1; 1 0 1 0; 1 0 0 1].](/images/equations/RedhefferMatrix/NumberedEquation1.svg)
Redheffer matrix is equal to the Mertens
function 
.
For 
,
2, ..., the first few values are therefore 1, 0, 
, 
, 
, 
, 
, 
, 
, ... (OEIS A002321).
of the eigenvalue 1 of the
Redheffer matrix is
(Vaughan 1993, p. 286; Vaughan 1996; Trott 2004, p. 57), where 
denotes the binary logarithm
and 
is the floor function. For 
, 
. The first few values for 
, 2, ... are 1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11,
... (OEIS A083058).
