Questions tagged [dirichlet-series]
For questions on Dirichlet series.
545
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Limit of the ratio of series
Prove or disprove the existence of a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following three properties are satisfied
$f(n) \leq n$ for each n;
The limit $\lim_{n\rightarrow \...
3
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Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,343
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How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?
If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are ...
- 1,725
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How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
- 759
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1
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Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
1
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55
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Asymptotic order of the square of the modulus of the second derivative of the Dirichlet kernel in zero
Consider the Dirichlet kernel $D_N(x)=\sum_{|k|\le N} e^{ikx}$. Its second derivative reads as $$D_N^{\prime\prime}(x) = -\sum_{|k|\le N} e^{ikx}k^2.$$
What is the asymptotic order of $|D_N^{\prime\...
- 11
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Dirichlet series solution to Poisson Point Process question
Reposted to MathOverflow because the bounty on this post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an edge ...
- 2,243
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About the definition of generalized harmonic numbers and an identity
Some software packages make use of the following definition for generalized harmonic numbers. In what follows, $\sigma,t\in\mathbb{R}$:
$$H_{ t }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \
(\sigma+it, t ...
- 363
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LCM sum with $\log $'s
If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
- 1,528
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can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
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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) ...
- 1,126
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The evaluation of the coefficient of the Dirichlet series $\zeta'(s)^2$
The derivative of Riemann zeta function is $\zeta'(s)=-\sum_{n=2}^{\infty}(\log{n}) n^{-s}.$
The square of $\zeta'$ is the following Dirichlet series:
$$\zeta'(s)^2=\sum_{n=4}^{\infty}a_nn^{-s},$$
...
- 962
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Can we substitute values after analytic continuation
Motivating example:
Consider the following $f(z)=\sum_{n\geq 0}A_n(z)$, and $g(z)=\sum_{n\geq 0} B_n(z)$.
Lets say $A_n(z)=\frac{(-1)^n}{(2n+1)^z}$ and $ B_n(z)=\frac{(-1)^{n}}{(n+1)^z}$
Then $f(z)=\...
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Why are the singular values equal to the first partial derivatives.
I am studying computer science so please go easy on me.
I am also too bad at math to extract the mathematical essence that is needed to answer this question so I'm just gonna explain the whole setup.
...
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Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
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inequality involving two dirichlet series
Let $f\left( s \right) = \sum\limits_{n = 1}^{\infty}\left[ a_{n} \cdot \left( {\frac{1}{n^{s}}-\frac{1}{n^{1 - \operatorname{conj}\left( s \right)}}} \right) \right]$ and Let $g\left( s \right) = \...
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How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.
Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
- 905
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Convolution Method for Bound
I am reading A survey of gcd-sum functions where the following result is stated:
Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by:
$$\sum_{n=1}^\infty \frac{P(...
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Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$
The paper mentioned a proposition:
$$
\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38}
K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }.
$$
Its equivalent is
$$
\int_{0}^{\infty}\vartheta_2(...
- 4,107
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Gibbs sampling equation for LDA
I don't understand what the w' means in the last derivation.
Here is the link for the paper: https://coli-saar.github.io/cl19/materials/darling-lda.pdf
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
A copy of this question is already montioned here in MO.
Note:I suggest adding Landau-Siegel zero tag here on SE
Using the following definition of Dirichlet L-function $$L(1,\chi)=\begin{cases}
\...
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Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$
I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$):
\begin{aligned}
\int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4
\text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
- 4,107
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Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$
The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function
$$\eta(s)=\underset{K\to\infty}{\text{lim}...
- 6,613
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What’s the best bound on the Dirichlet coefficients of $\zeta(s-1)^2/\zeta(s)$
We have $\frac{\zeta(s-1)^2}{\zeta(s)} = \sum\limits_{n\ge 1} \frac{a_n}{n^s}$, where $a_n = \sum\limits_{d|n} \mu(d) \sigma_0(\frac{n}{d}) \frac{n}{d} = \sum\limits_{d|n} \phi(d) \frac{n}{d}$. Here $\...
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What is the value of $L'(1,\chi)$ where $\chi$ is the non-principal Dirichlet character modulo 4?
I was trying to compute the following sum:
$$\sum_{n\le x}{\frac{r_2(n)}{n}}$$
where $r_2(n)=\vert\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\vert$. Using Abel's summation formula with $a_n=r_2(n)$, $\varphi(t)...
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Proof that ring of formal Dirichlet series is isomorphic to a ring of formal power series over countably many variables
I found this article of E.D. Cashwell and C.J. Everett "The ring of number-theoretic functions" and they said Dirichlet series ring is isomorphic to formal power series ring of countably ...
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Given a Dirichlet series that diverges, are there conditions to know when the modulus goes off to infinity?
I was working on a problem, and I had made the assumption that given a Dirichlet series
$$
L(s,f)=\sum_{n\geq 1}\frac{f(n)}{n^s}
$$
If I have some $\sigma\in\mathbb{C}$ such that $L(\sigma,f)$ ...
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Dirichlet series for $\frac{\zeta(1-s)}{\zeta(s)}$ [closed]
Wikipedia (here) says that $\frac{\zeta(s-1)}{\zeta(s)}= \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}$ where $\varphi(n)$ is the totient function. Similarly, is there a known expression involving a ...
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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 $\...
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
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1
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
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Finding the sum of a series using a Fourier series
I am stuck on how to calculate the value of the following sum:
$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$
I am aware that you need to find the corresponding function whose Fourier series is represented ...
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How to compute constants in asymptotic density of numbers divisible by subset of primes
I'm interested in the asymptotic density of the set $S$ of natural numbers divisible only by primes $p \equiv 1 \bmod 4$ (and similar subsets of $\mathbb{N}$). I'm aware of results which show that the ...
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Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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Why does $\sum\limits_{n=1}^\infty \frac{\nu(n)}{n^s} = \sum\limits_{m=1}^\infty \frac{1}{m^s}\sum\limits_p \frac{1}{p^s}$ hold
In context of a exercise about expressing the dirichlet series $$\sum\limits_{n=1}^\infty \frac{\nu(n)}{n^s}$$ in term of the zeta function, where $\nu(n)$ denotes the amount of different prime ...
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
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How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
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Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$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$$ ...
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How to prove the following Dirichlet-series/geometric-series idenity, step by step process?
$$\frac{\zeta(s)}{\zeta(hs)} =\prod_p\left(\frac{1-\frac{1}{p^{hs}}}{1-\frac{1}{p^{s}}}\right) =\prod_p\left(1+\frac{1}{p^s}+\cdots +\frac{1}{p^{(h-1)s}}\right)=\sum_{n\in S_h}\frac{1}{n^s}$$
What is ...
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Can $\alpha$ be found for $\sum_{n=1}^{\infty}\frac{\sigma_0(n^2)}{\sigma_0(n)}\frac{1}{n^s}=\zeta(s)\sum_{n=1}^{\infty}\frac{\mu^2(n)\alpha }{n^s}$?
I was looking for a pattern among these below:
$$ \sum_{n=1}^{\infty} \frac{\sigma_0(n^2)}{n^s} = \zeta^2(s) \sum_{n=1}^{\infty} \frac{ \mu^2(n)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)} $$
$$ \sum_{n=1}^{...
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Laplacian Dirichlet eigenvalues on a given domain
Let $\Sigma=[-1,1]\times[0,1]\cup[0,1]\times[-1,1]$ be an L-shape domain, over which I'm solving the Laplacian equation with Dirichlet boundary condition $$-\Delta f=\lambda f$$
I try applying the way ...
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example on a periodic signal that have infinite number of discontinuities in one fundamental period only?
I had a question on can a periodic signal have infinite number of discontinuities in one fundamental period only ?
the answer is yes.
but I wanted examples on it
another question was: example on a ...
6
votes
2
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Write the sum $\sum\limits_{a \in \mathbb{N}}\sum\limits_{b \in \mathbb{N}} \frac{(a,b)}{a^sb^t}$ in terms of the Riemann zeta function
I have the following exercise, and I need some help:
Write the sum
$$\sum\limits_{a \in \mathbb{N}}\sum\limits_{b \in \mathbb{N}} \frac{(a,b)}{a^sb^t}$$ in terms of the Riemann zeta function ($(a,b)$ ...
- 81
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How to prove $1-\frac{1}{2^x}\lt\eta (x)\lt 1$?
Define
$$\eta (x)=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}.$$
I managed to prove that
$$\left|\eta (x)-\left(1-\frac{1}{2^x}\right)\right|\le\frac{1}{2^x}$$
for $x\gt 1$ by the alternating series ...
- 407
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0
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53
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Dirichlet's Series - Convergence
Calculate the expression of the following Dirichlet's series:
$$ \dfrac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \dfrac{\varphi(n)}{n^s} $$
$$ \dfrac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty} \dfrac{...
2
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1
answer
87
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How to find the sum of this infinite series
I am not sure how to evaluate the infinite sum:
$$\sum_{n=0}^\infty \frac{1}{(2n+1)^6}$$
Apparently, I can shift it to
$$\sum_{n=1}^\infty \frac{1}{(2n-1)^6}$$
which is supposed to be a well known sum ...
- 21
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0
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62
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Question on convergence of product and Dirichlet series representations of a function
Consider the following two representations of $f(s)$
$$f(s)=
\underset{K\to\infty}{\text{lim}}\left(\prod\limits_{k=1}^K \left(1-\frac{2}{\left.p_k\right.^s}\right)\right)\tag{1}$$
$$f(s)=\underset{N\...
- 6,613
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Uniformly convergent series manipulation
I get confused reading about L-series and there is a lemma on infinite series. The question should only concern about analysis and there should be no number theory involved. The lemma is below,
Let $\{...
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1
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$\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$ and Riemann Zeta function
Is it possible to write $\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$, where $\Lambda(n)$ is the Von Mangoldt function and $\varphi(n)$ is the Euler totient function, ...
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1
answer
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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 ...
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