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March 22

Value of an L-function

I'm interested in the s-derivative at s=0 of the analytic continuation of

\mu (s)=\sum _{z\in \mathbb {Z} [i]}{\frac {}{}}\!\!^{\prime }{\frac {1}{|z|^{2s}}}.

Up to an overall gamma function and a constant, by taking the proper Mellin transform this is equal to the integral

\mu (s)=\int _{0}^{\infty }\theta (x)^{2}x^{s-1}\,dx,

where θ is a Jacobian theta function. Does this function have a name? (The Hurwitz zeta function initially seems promising, but doesn't quite do the trick.) I'd like to compute $\mu '(0)$ , provided it can be done in elementary transcendental functions. 71.182.216.55 (talk) 01:29, 22 March 2009 (UTC)[reply]

What is

\theta

here? Algebraist 01:32, 22 March 2009 (UTC)[reply]

Sorry, I made some edits in the mean time. 71.182.216.55 (talk) 01:35, 22 March 2009 (UTC)[reply]

Anyway, up to constants,

\theta (x)=\sum _{n}e^{-n^{2}x}

. 71.182.216.55 (talk) 01:40, 22 March 2009 (UTC)[reply]

Yeah, so the question is still live, despite recent activity on earlier threads ;-) Is anyone here knowledgable about arithmetic? 71.182.216.55 (talk) 04:08, 22 March 2009 (UTC)[reply]

The problem is interesting because it is the zeta function regularization of the determinant of the Laplacian on the flat (square) torus. Despite the fact that the should be the easiest test case of the determinant, it actually appears to me to be non-trivial. (It turns out that disks in symmetric spaces are easier, provided one believes in radial functions.) 71.182.216.55 (talk) 04:43, 22 March 2009 (UTC)[reply]

Birthdays

I am wondering: (A) if there are any actual studies or "real" data to answer my question or, if not, (B) if anyone has any relevant ideas or theories to suggest. My question is this. Statistically speaking, is any one day of the year equally likely to be someone's birthday as any other day of the year? In other words ... does a birthday of January 1 come up with equal probability as a birthday of January 2, January 3, January 4, ... and so on ... until December 31? Perhaps stated another way ... does each day of the year actually have a probability of 1/365? (I assume that, at least theoretically, they do ... right?) Are there any data or studies about this? If not, can anyone think of any ideas / reasons / theories as to why one particular birthday might show up with greater (or lesser) frequency than another? For sake of simplicity and convenience in this question, let's ignore the birthday of February 29. Thanks. (Joseph A. Spadaro (talk) 20:15, 22 March 2009 (UTC))[reply]

Google gave me this which, concludes, among other things, that conception is more likely to occur in winter and less in summer. That's for people alive in a specific period in a single country, so it may not reflect general trends accurately. The sample also contains people born in many different years, which smooths out effects like the fact (I don't know if this has a significant effect or not) that doctors are less likely to carry out caesareans and such at weekends. Algebraist 20:32, 22 March 2009 (UTC)[reply]

Thanks. Yes, I considered both of those issues. First, "winter" in one half of the world is "summer" in the other half of the world ... so the summer/winter distinction should not affect birthday statistics. Also, for example, August 17 might fall on a weekend in one year, but on a weekday in a different year ... so the weekend/weekday distinction also should not affect statistics. I believe? Thanks. (Joseph A. Spadaro (talk) 20:58, 22 March 2009 (UTC))[reply]

The world population is heavily hemispherically-skewed, so the seasonal effects will not obviously balance out. Different cultures and climates might have different seasonal effects, though. Algebraist 21:01, 22 March 2009 (UTC)[reply]

And the population of the southern hemisphere is heavily skewed towards the equator. There are very few (I think, almost no) people living more than 45 degrees south, compared to a very large number living more than 45 degrees north. --Tango (talk) 21:20, 22 March 2009 (UTC)[reply]

In some cultures you might get people aiming to give birth on certain days, and avoid others. They could therefore take various actions to move the birth date of their child, or perhaps lie about it if they failed to move it. StuRat (talk) 03:18, 23 March 2009 (UTC)[reply]