It seems there is a nice “pattern” between the continued fractions for the *Riemann zeta function*,

at *s* = 2, and *s* = 3. First though, a little introduction.

The origins of this function go back to 1644 when, at the tender age of 18, the Italian mathematician Pietro Mengoli (1626-1686) first proposed what would be later known as the *Basel Problem*, namely to determine the exact value of the sum of the reciprocals of the squares. Euler would later find that, for *n* a positive integer, then is a rational multiple of , with the first case *s* = 2 as,

which obviously is irrational. However, the status of odd *s* was not as easy to resolve. It was only in 1979 that Apery proved that is irrational, which henceforth was called *Apery’s constant*.

**I. Zeta(2)**

One of its infinite number of continued fractions can be given as,

where and the , starting with *n* = 1, are generated by,

or simply the odd numbers. The convergence is slow, but Apery found it can accelerated by using a quadratic function,

but now .

**II. Zeta(3)**

Likewise, this also has an infinite number of continued fractions (see Ramanujan’s versions here), but one important form is,

where and the , again starting with *n* = 1, are,

Apery again found an accelerated version,

where now , and established that its rate of convergence was such that could not be a ratio of two integers.

**III. Connection between Zeta(2) and Zeta(3)**

Define the two sequences,

then it was established that these *Apery numbers* have the *recurrence relations*,

Interesting that the *same polynomials* pop up, isnt’ it? Furthermore, their limiting ratios have the common form,

thus for *b* = 1, 2,

hence the *golden ratio* and the *silver ratio* surprisingly turn up in the continued fractions for and , respectively. It is easy to check other sequences,

for some small {*p, q*}, but there are no other recurrence relations similar in form to the two above.

**IV. Zeta(5)**

There *is* an orderly non-simple continued fraction for all , with the next odd *s* as,

where and the are,

Unfortunately, no one has yet found an accelerated version where the are generated by a 5th degree (or higher) polynomial, and *m* is rational.

*Can you find one?*

For further reading, refer to Alfred van der Poorten’s excellent, *A Proof That Euler Missed*.

Posted by A new continued fraction for Zeta(3)? « tpiezas on May 4, 2012 at 6:35 pm

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