TOPICS

Redheffer Matrix

A Redheffer matrix is a square -matrix with elements equal to 1 if or ( divides ), and 0 otherwise. 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).

The number of unit eigenvalues of the Redheffer matrix for is equal to

(Vaughan 1993, 1996; Trott 2004, p. 57), giving the first few values as 1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, ... (OEIS A083058).

(0,1)-Matrix, Mertens Function

Explore with Wolfram|Alpha

More things to try:

References

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. http://www.mathematicaguidebooks.org/.Vaughan, R. C. "On the Eigenvalues of Redheffer's Matrix. I." In Number Theory with an Emphasis on the Markov Spectrum (Provo, UT, 1991) (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.

Redheffer Matrix

Cite this as:

Weisstein, Eric W. "Redheffer Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RedhefferMatrix.html