All Questions
Tagged with dirichlet-series convergence-divergence
49
questions
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A question about Lemma 15.1 (Landau’s theorem for integrals) in Montgomery-Vaughan’s book
Lemma 15.1 in Montgomery-Vaughan’s analytic number theory book is Landau’s theorem for integrals. My question is, why is it necessary to have $A(x)$ bounded on every interval $[1,X]$? Doesn’t the ...
- 405
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1
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44
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For what values of $c$ is $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0$ when $x=\exp \left(-\frac{\rho _1}{c}\right)$?
The alternating Dirichlet series, the Dirichlet eta function, can be written in the form: $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}$
For what values of $c$ is $$\sum _{k=1}^{\infty } (-1)^{k+1}...
- 7,448
1
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1
answer
81
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Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$
Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$
is a sequence of positive real numbers that
$$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$
Does exists some $\epsilon>0$ such that $\...
- 33
1
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51
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Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$
Let: $$s=\rho _1$$ ...
- 7,448
4
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1
answer
109
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Abscissa of convergence of a Dirichlet series with bounded coefficients and analytic continuation [closed]
If a Dirichlet series has coefficients +1 and -1 and an analytic continuation without poles (or zeros) to the right of Re(s) = 1/2, what can we say about it's abscissa of convergence?
Is it always at ...
- 41
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266
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Proof Riemann Zeta Series based on $\eta(s)$ has only one pole.
This proof is my understanding of a very interesting comment by @leoli1 on my previous related question about the following extended Riemann Zeta function which converges for $\sigma>0$ where $s=\...
- 3,325
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143
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What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?
What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?
In the lecture note our instructor claimed that the abscissa of convergence of the above ...
- 211
1
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60
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Summary of Dirichlet Series Convergence from Apostol's IANT
I've been trying to learn about Dirichlet series, in particular from Apostol's IANT textbook.
The textbooks tend to present result and not discuss them narratively, so I am left unsure of my correct ...
- 3,325
1
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1
answer
474
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For which $s$ does $\sum 1/p^s$ converge?
A well-known result is that $\sum 1/n^s$ converges for $\operatorname{Re}(s)>1$.
Question: For which $s$ does $\sum 1/p^s$ converge, where $p$ is over all primes?
Notes:
Intuitively there are ...
- 3,325
2
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1
answer
183
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Confused by boundedness and convergence of Dirichlet Series (Apostol 11.6 Lemma 2, Theorem 11.8)
Apostol's IANT Section 11.6 is on "The half-plane of convergence of a Dirichlet Series".
In it he proves that if a Dirichlet series
is bounded at $s_0$ then it is also bounded at $\sigma>...
- 3,325
1
vote
1
answer
159
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Validity of proof showing difference in abscissa of convergence and absolute convergence of Dirichlet Series is $\leq1$?
The following is a step-by-step proof/derivation showing the difference in abscissae of convergence $a_c$ and absolute convergence $a_a$ is never more then 1.
Question: Is this simple proof correct?
...
- 3,325
0
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0
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51
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Why is $\lim_{x\rightarrow \infty} \sum_{x<n\leq\infty}a_{n}n^{-s}= 0$ a sufficient condition for convergence?
Assume the following is true
$$\left|\sum_{x_{1}<n\leq x_{2}}\frac{a_{n}}{n^{s}}\right| \leq Kx_{1}^{-\sigma}$$
where $s=\sigma+it$ and $a_n$ are complex, and all other variables are real, and $\...
- 3,325
0
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229
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Why is Abel's Identity (Apostol Theorem 4.2) valid for complex functions?
Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below)
$$ \sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt $$
to prove a result about complex ...
- 3,325
0
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0
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74
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Is this argument sufficient to show alternating zeta series converges for $Re(s)>0$?
The alternating zeta series
$$ \eta(s) = \sum\frac{(-1)^{n+1}}{n^s} $$
is known to converge for $\sigma=\Re(s)>0$.
Question: There are many proofs offered but I wondered if the following simple (...
- 3,325
2
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0
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80
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Simple argument to explain why the series for $\zeta(s)$ diverges for Re(s)=0
Question
I am trying to find a simple argument students not trained to university level maths that the series
$$\zeta(s) = \sum \frac{1}{n^s}$$
diverges for $\sigma=0$, where $s=\sigma+it$.
Here "...
- 3,325