Since we're on the topic of Bernoulli numbers, Barry Mazur's lectures on their universality is worth reading: http://wstein.org/wiki/attachments/2008%282f%29480a/Bernoulli.pdf On Wed, Jan 7, 2015 at 4:09 PM, Victor Miller <victorsmiller@gmail.com> wrote:
I wonder if this is related to the "standard" extension of Bernoulli number to have p-adic index. First, if zeta denotes the Riemann zeta function and k is an integer >=2 we have
zeta(1-k) = -B_k/k
In the p-adic integers, the ordinary positive integers are dense. The Kummer congruence shows that -B_k/k is p-adically continuous,so there is a unique p-adic continuous function which interpolates these values. It turns out to be p-adic analytic. Here's a nice survey about it: http://www.rac.es/ficheros/doc/00261.pdf
Victor
On Wed, Jan 7, 2015 at 3:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Turns out I still had typos. Trying again. Goal is to devise a function Bernoulli(n) defined for complex n (not nec'ly integers) and such that reals map to reals, and agrees with Bernulli numbers when n=0,1,2,3,... http://en.wikipedia.org/wiki/Bernoulli_number
My original suggestion with typos hopefully now finally corrected, using a ln(z) definition slit along the positive real axis (which is not the standard definition), and A^B = exp(B*ln(A)), was
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral z^(-n) / [exp(z)-1] dz
using an anticlockwise contour of integration like x = y^2 - 1. However, saying that about the nonstandard ln was somewhat cavalier of me. If we use the standard ln(z) definition with slit along the negative real axis, then
Bernoulli(n) = Gamma(n+1)/(4*pi*i) * integral [ exp(n*[pi*i-ln(-z)]) + exp(n*[-pi*i-ln(-z)]) ] / [exp(z)-1] dz
would work where the average of the two exp's (and the reflection symmetry of the contour) serves to cancel out imaginary parts so that Bernoulli(n) is real if n is real.
However, Asimov may wish to reconsider his desire that it map real-->real. To explain, consider the functions sin(x) and cos(x), which both map real-->real. Do you not think exp(i*x) is somehow more fundamental? If so you might prefer something like
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral exp(n*[pi*i-ln(-z)]) / [exp(z)-1] dz
which does not map reals to reals, but its real part is my previous definition...
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