29
votes
Accepted
1+2+3+4+… and −⅛
Yes there is, in $p$-adics.
You are probably familiar with the relation
$$8T(n)+1=(2n+1)^2.$$
Now for any $p$ except $2$ (which has to be excluded because of the non-unit coefficients in the above ...
- 2,370
17
votes
Accepted
Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$
The sum is equal to
$$
\int\limits_{0}^{\infty}\frac{\exp(-x)}{1+W_0(x)}\,\mathrm{d}x = 0.7041699604...
$$
where $W_0(x)$ is the Lambert-$W$ function.
Reference
Stephen Finch. "Errata and Addenda ...
- 2,826
16
votes
1+2+3+4+… and −⅛
You are not using all partial sums, but only a restricted choice of them. So you are not really looking at the limit of all partial sums. An analogue would be deciding to compute $1 - 1 + 1 - 1 + 1 - ...
- 50.6k
13
votes
Accepted
Divergent series summation beyond natural boundaries
From those three examples, Rogers-Ramanujan's series belong to the class of basic hypergeometric series ($q$-series). It is a marginally logarithmic divergent series for $q > 1$ and it should be ...
- 2,826
10
votes
On modified Euler product
Expanding as a power series in $x$
$$\log(1-x) = -x - \frac{x^2}2 -\frac{x^3}3 - \cdots $$
Thus
\begin{eqnarray*}
\log F(s) &=& \sum_p -\log(p)\log (1-c p^{-s}) \\
&=& \sum_p \log(p) \...
- 1,243
10
votes
Accepted
Is the pseudoinverse the same as least squares with regularization?
Here's one way to see this directly. I will assume that $A$ is $m \times n$. Let
$$ A = U \Sigma V^T$$
be the SVD for A. Recall that the regularized solution to the least squares problem $Ax = b$ is ...
- 2,607
10
votes
Is the pseudoinverse the same as least squares with regularization?
Yes, they are connected.
The first problem is a special case of the second when $\lambda=0$, if the first problem has a unique solution (i.e., $\ker A = 0$), or if the second is also formulated as a ...
- 20.2k
9
votes
Is regularization of infinite sums by analytic continuation unique?
Let $$f_k(s) = k^{-s}+(s+1)k^{-s-2},\qquad f_k(-1)=k$$ then $$F(s)=\sum_k f_k(s) = \zeta(s)+(s+1)\zeta(s+2), \qquad F(-1)=-1/12+1$$
- 3,403
8
votes
Is the pseudoinverse the same as least squares with regularization?
TL;DR : Yes, the two problems are equivalent in the limit $\lambda \rightarrow 0$ !
One freely available reference is the following (excellent in my opinion) review paper : https://arxiv.org/abs/1110....
- 297
8
votes
Accepted
Why we cannot speak about the main or natural regularization?
Preliminary comment: I second the idea that, for certain integrals, there should be a "well-behaved" class of regularization methods which all give the same value - of course, the important thing then ...
- 12.5k
6
votes
Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?
Two pieces of good news.
The formulae $\int_{0}^\infty \sin(x)\,dx=1$ and $\int_0^\infty \tan x\,dx=\ln 2$ hold. (Can‘t say anything about $\psi$ since I don‘t know what it means).
These are not ...
- 2,472
5
votes
Is regularization of infinite sums by analytic continuation unique?
As the other answer has pointed out, $-1/12$ is not the only value that can obtained with analytic continuation. However, it is the unique constant term of the asymptotic expansion of the smoothed ...
- 2,203
5
votes
Regularizing the sum of all primes
It was too long for a comment and overall as far as possible can be from rigorous so if it isn't helpful just tell me and I'll delete it.
In his blog John Baez talks of particular approach that he ...
- 256
5
votes
Comparing sizes of sets of natural numbers
Your post expresses the same idea about comparison of infinite sets as in my previous post with the difference being that you use Abel summation while I did use Ramanujan's summation and Zeta ...
- 10.1k
5
votes
Less fundamental applications of Zeta regularization:
Zeta-function regularization of the determinant of the Laplacian, for example on a torus, might qualify as a "purely mathematical" application.
See, for example, On functional determinants of ...
- 188k
5
votes
Accepted
Derivative of Cauchy PV is equivalent to Hadamard regularization?
A derivation of the relation
$${\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {f(t)}{t-x}}\,\mathrm dt\right)={\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,\...
- 188k
4
votes
Less fundamental applications of Zeta regularization:
Zeta function regularization computes the asymptotics of smoothed sums.
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-...
- 4,217
4
votes
Does this method analytically continue gap series series?
For $k=1$ your formula indeed gives an analytic continuation, but for $n\geq 3$, it is known that your function $f$ has no analytic continuation (the unit circle is the natural boundary of your ...
- 91.8k
4
votes
A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Summation defined at non-natural values is also known as "fractional summation" (even if sum limits belong to ℂ). Markus Müller and Dierk Schleicher have provided a proper axiomatic ...
- 2,826
3
votes
Theta-function in the lower half-plane
If you are still active, I would be very interested to see the finite values you obtain in a physical context. For now, however, I will present two equal approaches we could take to regularize the sum....
- 1,730
3
votes
Is the pseudoinverse the same as least squares with regularization?
As usual with such problems, it is most insightful to forget about matrices for a while and think about abstract vector spaces instead.
Let $V$ and $W$ vector spaces and $A: V\to W$ linear†. ...
- 213
3
votes
Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?
Well, after some thinking, I came to the following method.
To find $\int_0^\infty \tan x\, dx$ we have to subtract from the mean value of the antiderivative at infinity its value at $0$. This follows ...
- 10.1k
3
votes
Accepted
Interesting questions for inverse parabolic problems
Inverse Problems for Partial Differential Equations (third edition, 2017) by Victor Isakov concludes each chapter with a collection of open research problems. Chapter 9 is specifically devoted to ...
- 188k
3
votes
Accepted
How to derive the solution of Tikhonov Regularization via SVD
No need for the Woodbury identity. Just replace $I$ with $VV^H$, and factor out the $V$s.
- 20.2k
3
votes
Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
As I said in the comment, $$F_a(s) = \sum_{n=0}^\infty (n^2+a)^{-s}, \qquad Re(s) > 1/2$$
Has an analytic continuation in term of the Riemann zeta function :
$$F_a(s) = \sum_{n= 0}^{A-1} (n^2+a)^{...
- 3,403
3
votes
Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
The sum is only convergent for $p>1/2$, so for $p=1/2$ you could regularize it by adding a small positive increment: $p=1/2+\epsilon$, $\epsilon>0$.
For $a\gg 1$ you can then approximate the ...
- 188k
3
votes
Comparing sizes of sets of natural numbers
Well, lots of time have passed and now I have an explicit formula for numerosity. It gives the same (up to an infinitesimal) differences between numerosities of sets as your formula, but can express ...
- 10.1k
2
votes
Regularized linear vs. RKHS-regression
Both of the penalties can be thought of as arising from the linear regression setting in a Bayesian framework with predictor matrix $K$ and a Gaussian prior over the vector $\alpha$, centered at zero ...
- 2,791
2
votes
On modified Euler product
I'll expand on Ralph's answer to describe how to evaluate $F(1/2)$.
Ralph's main point is that $\log F(s)$ is well-approximated by a sum of logarithmic derivatives of $\zeta(s)$. Writing it out ...
- 1,570
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