# Zeta regulated product, solving without the zeta function.

Earlier i've asked about how to calculate divergent products, i got some directions which made me curious. Now i'm wondering is this correctly done.

Divergent products.

The most commen divergent product, and also mentioned in the topic quite often was: $$\prod_{n=1}^{\infty}(n)= (2\pi)^{0.5}$$ So i wanted to check it myself.

I used the following formula.

$$\sum_{n=1}^{dp} f(n)\sum_{k=1}^{d-1} (e^{\frac{2i\pi k}{d}})^{n}=\sum^p_{n=1} (d f(nd)-f(n))$$

With d an integer (the period) and d>=2, and p tends to infinity.

I got this formula the following way:

$$\sum_{n=1}^{dp} \sum_{z=0}^{d-1}f(n+z)\sum_{k=0}^{d-1} (e^{\frac{2i\pi k}{d}})^{n+z}=\sum^p_{n=1} d^2 f(nd)$$

And after corrections, this gives : $$\sum_{n=1}^{dp} \sum_{z=0}^{d-1}f(n+z)\sum_{k=1}^{d-1} (e^{\frac{2i\pi k}{d}})^{n+z}-\sum^{d-1}_{z=1}(d-z)f(z)=\sum^p_{n=1} (d^2 f(nd)-df(n))$$

Which give: $$\sum_{n=1}^{dp} f(n)\sum_{k=1}^{d-1} (e^{\frac{2i\pi k}{d}})^{n}=\sum^p_{n=1} (d f(nd)-f(n))$$

I used this also to calculate the zeta funtions for example. There are some function were this won't work, or need to be correct because of constants, so question 1, what is the "real" mathematical formula to transform (divergent) sums into it's alternating form.

$$\prod_{n=1}^{\infty}(n)= e^{\sum_{n=1}^{\infty} ln(n)}$$ $$\sum_{n=1}^{\infty} ln(n)= \sum_{n=1}^{\infty} 2ln(2n/4)-ln(1/4) =\sum_{n=1}^{\infty} (-1)^n*ln(n/4)$$

$$\sum_{n=1}^{\infty} -(-1)^n*ln(4)=1/2 (ln(4))$$

So i'm left with $$\sum_{n=1}^{\infty} (-1)^n*ln(n)=$$ $$1/2(\sum_{n=1}^{1/2(m)} ln(2n)-\sum_{n=1}^{1/2(m)} ln(2n-1) + \sum_{n=1}^{1/2(m-1)} ln(2n) -\sum_{n=1}^{1/2(m+1)} ln(2n-1) )=$$ $$(2m-1)/2*ln(2)+ln((1/2*m)!)+ln((1/2(m-1))!)-ln(m!)=$$ $$-1/2ln(2)+ln(\frac{((1/2)m)!*((1/2*(m-1))!)*2^m}{m!})=1/2(ln(pi/2))$$

Which solved the problem: $$\prod_{n=1}^{\infty}(n)= (2\pi)^{0.5}$$

Because i'm no mathematician, everything i did is done on intuition, is this correctly done? And i have troubles solving the following sum, if i want to use the same idea, how should i solve: $$\sum_{n=1}^{\infty} ln(n+1)$$.

• The product of interest does not converge. Commented Jun 14, 2015 at 16:57
• It's why i said divergent products and sums in the title. Commented Jun 14, 2015 at 18:16
• Why did you write that the divergent product equals $\sqrt{2\pi}$? Commented Jun 14, 2015 at 18:27
• Why i did write it down in the first place, was that it was stated in the previous topic as derivative value of zeta(o). Since it's hard, atleast for me, to find the derivative of the zeta function, i tried if there was an easier way to calculate it. So when i did it myself (as shown above) i get the same value. Why i assign a finite value to a divergent product is because i belief it's the constant finite part. Commented Jun 14, 2015 at 19:16
• @MarkViola $\prod_n n = \exp \sum_n n^{-s} \log n |_{s = 0} = \exp \left(-\sum_n \frac{d}{ds} n^{-s} |_{s = 0}\right) = \exp \left(-\zeta'(s)|_{s=0}\right) = \exp \left(-\zeta'(0)\right) = \sqrt{2\pi}$. Commented Mar 5, 2020 at 19:46