- algorithm Apollonian approximation art bar beauty binary binomial coefficients book cards carnival Carnival of Mathematics chocolate circle complex counting decadic decimal diagrams elements factorization fibonacci formula fractal game games gasket graph groups Haskell hyperbinary idempotent integers interactive irrational Ivan Niven Lagrange lehmer lucas MaBloWriMo making Mersenne moebius mu multiplication nim number numbers objects omega order paper pi prime primitive programming proof puzzle rectangles review roots sequence square strategy subgroups sum symmetry tessellation test triangular unit unity video visualization X
### Blogroll

### Fun

### Reference

### Categories

- algebra (43)
- arithmetic (59)
- books (28)
- calculus (6)
- challenges (51)
- combinatorics (12)
- complex numbers (6)
- computation (42)
- convergence (9)
- counting (31)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (24)
- geometry (56)
- golden ratio (8)
- group theory (26)
- humor (6)
- induction (7)
- infinity (17)
- iteration (23)
- links (74)
- logic (6)
- meta (40)
- modular arithmetic (24)
- number theory (68)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (81)
- people (20)
- pictures (59)
- posts without words (15)
- primes (34)
- probability (6)
- programming (17)
- proof (66)
- puzzles (11)
- recursion (12)
- review (19)
- sequences (28)
- solutions (28)
- teaching (13)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- January 2017 (2)
- December 2016 (4)
- November 2016 (6)
- October 2016 (6)
- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

### Meta

# Author Archives: Brent

## More fun with Dirichlet convolution

I’m back after a bit of a hiatus for the holidays! Last time we saw how the principle of Möbius inversion arises from considering the function from the point of view of Dirichlet convolution. Put simply, the Möbius function is … Continue reading

Posted in number theory
Tagged arithmetic, convolution, Dirichlet, divisors, inversion, moebius, mu
Leave a comment

## Paper torus with Villarceau circles

I made another thing! This is a torus, made from 24 crescent-shaped pieces of paper with slots cut into them so they interlock with each other. I followed these instructions on cutoutfoldup.com. There is also a template with some ideas … Continue reading

## Möbius inversion

In my last post we saw that , that is, the Möbius function is the inverse of with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion. Möbius inversion. Suppose is defined for as the … Continue reading

## Dirichlet convolution and the Möbius function

Recall from last time that the Dirichlet convolution of two functions and is written and defined by: where the sum is taken over all possible factorizations of into a product of positive integers. Last time we saw that is commutative … Continue reading

## Dirichlet convolution

Let and be two functions defined on the positive integers. Then the Dirichlet convolution of and , written , is another function on the positive integers, defined as follows: The sum is taken over all possible factorizations of into a … Continue reading

## The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof
Tagged circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity
2 Comments

## The Möbius function proof, part 1

In my last post, I introduced the Möbius function , which is defined in terms of the prime factorization of : if has any repeated prime factors, that is, if is divisible by a perfect square. Otherwise, if has distinct … Continue reading