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September 23

Universal algebra based on free logic

My question is about whether universal algebra, e.g. varieties, is generally based on free logic (which I understand to allow an empty universe of discourse) rather than first-order logic (which I understand to axiomatically deny such). Examples where this makes a difference would include magmas, semigroups and quasigroups, in the sense of whether the empty structure in each case would or would not be permitted. These articles suggest that these are permitted, from which I deduce that first-order logic is not the basis for their construction; in fact, it seems to me that many theorems about these structures would become convoluted with exceptions ("exceptionally convoluted"?) if the empty structure was not to be permitted. —Quondum 01:48, 23 September 2014 (UTC)[reply]

Trig sum

Does the sum

$\sum _{n=1}^{\infty }{\frac {\sin {\frac {1}{n}}}{n}}\approx 1.472828$

have a known closed form? 70.190.182.236 (talk) 07:03, 23 September 2014 (UTC)[reply]

http://www.wolframalpha.com/input/?i=sum_{n%3D1}^infinity+sin%281%2Fn%29%2Fn confirms the value 1.472828231868503, but gives no closed form. Bo Jacoby (talk) 09:42, 23 September 2014 (UTC).[reply]

If you expand the sine out in a series, this can be re-expressed as

\sum _{k=0}^{\infty }{\frac {(-1)^{k}\zeta (2k+2)}{(2k+1)!}}

where ζ is the Riemann zeta function. Its values at the positive even integers is known to be

\zeta (2k)={\frac {(-1)^{k+1}B_{2k}(2\pi )^{2k}}{2(2k)!}}

where B_2k is a Bernoulli number. So the summation reduces to

\sum _{k=0}^{\infty }{\frac {B_{2k+2}(2\pi )^{2k+2}}{2(2k+2)!(2k+1)!}}

Still not a closed form, but perhaps marginally better than what you started with. (This series also converges much more rapidly.) Sławomir Biały (talk) 11:55, 23 September 2014 (UTC)[reply]

This inverse symbolic calculator gives some matches for 1.472828, but none match the further digits given by Wolfram Alpha. Staecker (talk) 12:00, 23 September 2014 (UTC)[reply]

Hardy-Littlewood Function

According to at least two references through google (I googled 1.472828), that is H(1) where H(x) = $\sum _{n=1}^{\infty }{\frac {\sin {\frac {x}{n}}}{n}}$ . Most of what I've found on the Hardy Littlewood functions are from the standpoint of them being very slowly convergent. Check out https://www.cs.purdue.edu/homes/wxg/slidesHL.pdf , I don't know where the lecture is that goes with those powerpoint slides, but should give you a start.Naraht (talk) 17:57, 23 September 2014 (UTC)[reply]