All Questions
Tagged with dirichlet-series mellin-transform
15
questions
1
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1
answer
99
views
Perron's Formula with $\rm{si}$-Remainder
I'm studying the book `Multiplicative Number Theory I. Classical Theory' by Hugh L. Montgomery and Robert C. Vaughan, and I don't understant a step of the proof for Perron's Formula(in Section 5.1) ...
- 1,145
0
votes
0
answers
170
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Intuition behind theta function L-series correspondence
I know that we can analytically continue $L$-series by taking the Mellin transform of a sutiable theta function. Technically we need a transformation law for the theta function at $0$ and $\infty$ as ...
user427380
1
vote
1
answer
118
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Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$
On page 87 of Ingham's book: The Distribution Of Prime Numbers, the author asserts the following results, but does not give proof.
Let $a(x)$ be a bounded and integrable function over any finite ...
- 915
0
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1
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68
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Question on convergence of explicit fomulas for summatory functions related to Dirichlet series
Given the totient summatory function
$$\Phi(x)=\sum\limits_{n=1}^x\varphi(n)\tag{1}$$
and the related Dirichlet series
$$\frac{\zeta(s-1)}{\zeta(s)}=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{...
- 7,631
1
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1
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161
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Is there a valid explicit formula for $f(x)=\sum\limits_{n=1}^x \frac{1}{n}\sum\limits_{d|n} \mu(d)\,d$?
This question is related to the function $f(x)$ defined in (1) below where A023900(n) is the Dirichlet inverse of Euler totient function $\phi(n)$. I believe the related Dirichlet series illustrated ...
- 7,631
3
votes
0
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143
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Questions about the Dirichlet series $K(s)=\sum\limits_{p^k} p^{\,-k\,s}$
This question is related to Riemann's prime-power counting function $J(x)$, the fundamental prime-counting function $\pi(x)$, and the simple prime-power counting function $k(x)$ defined below where $p\...
- 7,631
2
votes
0
answers
113
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Question about Dirichlet Series Related to Formula for $\frac{1}{e}$
This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$.
(1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
- 7,631
0
votes
1
answer
58
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Existence of classical Dirichlet series from Lebesgue integrable inverse Mellin transform
Let $f(s)$ be meromorphic in $\mathbb{C}$. Let the following inverse Mellin transform be Lebesgue integrable for all real positive $x$ at some complex point $s$ with some real $c$:
$\frac{1}{2\pi i} \...
- 548
6
votes
0
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202
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Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$
This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$.
(1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$
(2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
- 7,631
1
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1
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179
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isometry relating $L^2(\mathbb{R}, dx)$ and $L^2([0, \infty), \frac{dx}{x})$
What is the basis for functions on the Hilbert space $L^2([0,\infty), \frac{dx}{x})$. I am studying the Mellin transform and I'm trying to understand the role of the functions $n^{it} = e^{it \, \log ...
- 24.5k
1
vote
1
answer
41
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Help me understand this theorem demonstration - Dirichlet series
On a chapter on Dirichlet series, there is this lemma:
Lemma: if $f \in C^\infty$ and $\forall n, k, |f^{(n)}(t)| =o(t^k)$ when $t \to +\infty $ then
$M(f, s) = \frac{1}{\Gamma(s)} \int_0^\infty f(...
- 1,124
1
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1
answer
139
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what is the name of this formula ? with mellin transform
$$ \sum_{n=1}^{\infty}a(n)f(n/x)= \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}G((s)F(s)\frac{ds}{x^{s}} $$
where $ G(s)= \sum_{n=1}^{\infty}a(n) n^{-s}$
and $ F(s)= \int_{0}^{\infty}dx x^{s-1}f(x) $
...
- 8,536
1
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1
answer
64
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On $\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\eta'(s)\frac{x^s}{s\zeta(s)}ds$, for $c>1$, where $\eta(s)$ is the Dirichlet Eta function
When I was combining the identities from this article from Wikipedia for the Mertens function, I've asked my an open question, if you can solve it from a standard viewpoint it is appreciated, and ...
user243301
2
votes
0
answers
104
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if $f(x)$ is periodic $\left|\int_1^\infty f(x) x^{-s} dx\right| \sim C\left|\int_1^\infty \sin(x) x^{-s} dx\right|$ when $\text{Im}(s) \to \infty$
is it true that
if $f(x)$ is periodic, non-constant and bounded $$\text{when } T \to \infty ,\qquad\qquad\sup_{|t| \ <\ T}\ \ \left|\ \int_1^\infty f(x) x^{-\sigma-it} dx\ \right| \ \sim \ ...
- 78.4k
1
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1
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2k
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Derivation of Perron's formula
I tried to derive Perron's formula, but got really screwed up. I know of other ways to derive it, but I'm not quite sure why this way isn't working. I would appreciate some pointers on where I'm going ...
- 8,000