Odd Sums of Consecutive Odds

Oscar Wilde’s character Algernon said in The Importance of Being Earnest, “One must be serious about something, if one is to have any amusement in life.” Of course in Wilde’s typical ironic fashion, Algernon was only referring to his own dedication to frivolous diversions. In that spirit, allow me a few moments to tell a story about one of the odder sums of odd integers I discovered as a kid.

I remember that sometimes when I was bored — most especially during long, bi-weekly car trips with my parents — I would play various games with integers. I have no idea why, but at one point I memorized some huge list of powers of 2 (I can still remember the list from 1 to 65,536). I also computed the squares, cubes, and so forth of most of the smaller integers. As a result, I discovered on my own quite a number of interesting patterns in the integers. I don’t remember most of them, but there is one in particular that has stuck with me through the years.

Adding, Multiplying and the Mellin Transform

    Matt’s entertaining posts on divergent series have inspired me to contribute my own two cents.  In his posts, the central theme has been to replace a divergent series with a formal ‘function’ and to use algebraic manipulations and analytic continuation to assign a ‘best’ sum to a series without any obvious candidate.  The two classes of formal functions used have been generating function and zeta functions .

    One of my favorite traits of these two beasts is the connections between them, in particular, an integral-based method for turning generating functions into zeta functions called the Mellin transform.  Why should such a connection make me happy?  Well, on one side, the generating function is useful for exploring the additive nature of the sequence of ‘s; while the zeta function is useful for exploring the multiplicative nature of the sequence of ‘s.  It is a sad fact of mathematical life that understanding the additive nature of an object (in say, a ring) is almost totally unhelpful for understanding the multiplicative nature of that object.  For instance, the prime factorization of an integer tells us virtually nothing about the prime factorization of (for a more concrete realization of the hardness of this problem, see the Collatz conjecture).   Therefore, the ability to turn a generating function into a zeta function can make impossibly obfuscated details clear!