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1
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1answer
221 views

Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
7
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0answers
132 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
4
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0answers
154 views

To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

This is a cross-post from MSE. In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees. The idea to construct such a ...
12
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1answer
276 views

Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
2
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1answer
105 views

Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values

Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form $S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ where $H_k = \sum_{n=1}^{k} ...
5
votes
1answer
295 views

What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?

Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider ...
8
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1answer
287 views

ζ(-n) and “powers” of Grandi's series

For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+...$ A value can also be assigned to the related series ${n+0 \choose n}+{n+1 ...
2
votes
1answer
282 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
4
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5answers
281 views

procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...
3
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1answer
370 views

What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question) I think I understood the concept of fractional derivatives applied to ...
3
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3answers
227 views

Asymptotic series

I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not ...
6
votes
1answer
330 views

Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
6
votes
1answer
357 views

Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m: $$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$ where I want to make ...
0
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1answer
188 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far] For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...
7
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2answers
1k views

Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$

UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text. This is a concise version of this math.SE question of mine. I've got an answer ...
14
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0answers
793 views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
5
votes
2answers
737 views

Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? I asked ...
9
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2answers
1k views

Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is: I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...
2
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1answer
379 views

Approximation:- Algorithmic considerations

Hello I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...
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2answers
2k views

Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$ See ...
9
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2answers
1k views

Abel summation of the alternating series of primes?

Consider the ordinary generating function of the sequence of primes ($2+3x+5x^2+7x^3 + ...$); by the ratio test and the prime number theorem, its radius of convergence is $1$. Thus, we might well ask ...
6
votes
1answer
1k views

Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?

(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.) The Ramanujan ...
4
votes
1answer
316 views

Is that series-transformation known in the context of divergent summation?

Note: I asked this question in math.stackexchange but did not receive an answer Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular ...
4
votes
1answer
459 views

mertens-function in the light of divergent summation - what summation method were best adapted

Just reading about the Mertens-function in the other thread Mertens function I remember an earlier attempt to apply divergent summation to the series which is constructed of the Moebius-function at ...
6
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3answers
813 views

Uses of Divergent Series and their summation-values in mathematics ?

This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see. Dear MO-Community, When I was trying to find closed-form representations for odd ...
24
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2answers
1k views

Is there a “quantum” Riemann zeta function?

Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the ...
1
vote
3answers
605 views

Closed form of divergent infinite product?

Okay, we know that $$ \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big) $$ . Is there some known (trigonometric(?)) function that is equal to the following infinite ...
7
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3answers
822 views

Summing a divergent series and a constant combined

At least according to the answer to this question, $\zeta(1) = \gamma $ (once reqularized, of course). Let me rephrase that by stating that: $$ \sigma(\zeta(1)) = \gamma $$ Here, $\sigma(x)$ is the ...
5
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3answers
389 views

Non-absolute convergence of series with asymtotically equal coefficients

The following seems to be a question related to standard calculus, but I am not quite sure where to look for an answer. Suppose $f,g:\mathbb{N} \to \mathbb{C}$ are such that the have the same ...
1
vote
1answer
415 views

Testing for asymptotic series

I would like to know if there is a systematic way of writing down an asymptotic series representation of a function? (like one can use Taylor series expansion for doing a power series). Conversely ...
12
votes
2answers
649 views

What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups?

The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be ...
4
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1answer
430 views

Values of cusp forms at q = 1 ?

Take a cusp form $f$ and let $f(q) = q + a_2q^2 + q_3q^3 + \ldots$" denote its $q$-expansion (assume that the $a_k$ are integers, and that $f$ comes from an elliptic curve $E$). Of course the series ...
23
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4answers
2k views

Do Abel summation and zeta summation always coincide?

This is a more focused version of Summation methods for divergent series. Let $a_n$ be a sequence of real numbers such that $\lim_{x \to 1^{-}} > \sum a_n x^n$ and $\lim_{s \to 0^{+}} > ...
18
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4answers
3k views

Summation methods for divergent series

There are many methods for assigning a value to a series that diverges, e.g. zeta function regularization, Abel summation, Cesaro summation, etc. From all of the examples I've found, two methods ...
26
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7answers
2k views

Does any method of summing divergent series work on the harmonic series?

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic ...