For questions on Dirichlet series.

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Dirichlet series with abscissa of absolute convergence $= \frac{1}{2}$

I'm trying to figure out a Dirichlet series which has its abscissa of absolute convergence $=\frac{1}{2}$. I've been trying to think about using the formula for this abscissa: ...
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52 views

Convergence of the Euler product

Suppose that the Riemann Hypothesis is true. It is well known that then the Dirichlet series $$\sum_{n=1}^\infty\frac{\mu(n)}{n^s}$$ converges in the half-plane ${\rm {Re}}\, s>\frac{1}{2}$. Does ...
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1answer
33 views

Green function for Dirichlet-problem for $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x)$

Let $G(x,\xi)$ be de Green function for the Dirichlet problem for $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x)$, where $p\in C^2$, $q \in C^1$, $r \in C^0$ and $p(x)\neq 0$ for every $x \in [a,b]$. ...
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Analytic Dirichlet series

I have another question regarding Dirichlet series. This one is about where the Dirichlet series $f(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$ is analytic. I have the next theorem: "$f(s)$ is ...
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1answer
41 views

We can suppose without loss… General Dirichlet question

I would like to know why I can start supposing that $\sigma_{0}=0$ in the proof of the following theorem about Dirichlet series: "Let $a_{n}\ge0$ for every $n\ge1$. Then the point $s=\sigma_{0}$ is a ...
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1answer
14 views

Are all series in the elementary Ramanujan class R = 1 non-summable by analytic continuation of Dirichlet series?

We say that a series $\sum_{n=1}^\infty a_n$ and the corresponding power series $f(x)=\sum_{n=1}^\infty a_nx^n$ belong to the Ramanujan class $R=1$ if $g(x)=f(x)-f(x^2)$ is Abel summable at $x=1$ ...
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101 views

Closed-form for $\sum_{n=2}^{\infty} \frac{1}{\pi (n)^2}$

Is there a closed form for $$\sum_{n=2}^{\infty} \frac{1}{\pi (n)^2}$$ (where $\pi (x)$ is the number of primes less than or equal to x)? It's obvious that $$\sum_{n=2}^{\infty} \frac{1}{\pi (n)^s}$$ ...
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23 views

the set of Dirichlet series converging for $Re(s) > \sigma$

$F(s) = \sum_{n=1}^\infty a(n) n^{-s}$ converging for $Re(s)$ large enough. let : $$A_1(x) = \sum_{n \le x} a(n), \qquad \qquad A_{k+1}(x) = \sum_{n \le x} A_k(x)$$ is it true that $\scriptstyle ...
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Dirichlet clustering and group assignment

As in the Dirichlet clustering, the dirichlet process can be represented by the following: Chinese Restaurant Process Stick Breaking Process Poly Urn Model For instance, if we consider ...
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1answer
52 views

Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?

If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
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21 views

A linear operator that extends the summation of Dirichlet series

Consider a vector space $\mathcal{V}$, a linear operator $L$ and a vector subspace $\mathcal{A}$ such that for all $x\in\mathcal{A}$ $Lx\in\mathcal{A}$ and for a number $R\neq0,1$ $R^{-1}$ is not a ...
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1answer
50 views

What's the function that is related to 3 as the Riemann zeta function is related to 2?

For $f(x)=\sum_{n=0}^\infty a_n x^n$, a real number $R\neq 1$, $g(x)=f(x)-Rf(x^2)$ Abel summable at $x=1$, $g(1)=\lim_{x\to 1^-} g(x)$, the elementary Ramanujan sum of $f(x)$ at $x=1$ is defined by ...
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61 views

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 \ ...
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1answer
44 views

Exploring the Dirichlet series of the sum of remainder function

I wolud like to learn and understand more some basic facts about Dirichlet series, for wich I want explore the following function, that is called the sum of remainders function, A004125 as Sloane's ...
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33 views

Is the limit of a Dirichlet series to infinity always equal to the constant term?

Given a sequence of complex numbers $(a_n)$ one defines the corresponding Dirichlet series as $$f(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s}.$$ I know that there is some $x_0 \in \mathbb{R} \cup \{\pm ...
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2answers
37 views

Determining the coefficients of the reciprocal of a Dirichlet series

Given a Dirichlet series with coefficients $$ F(s)= \sum_{n=1}^{\infty}\frac{a(n)}{n^{s}},$$ is it then always possible (and how) to obtain the $ b(n) $ coefficients related to its reciprocal $$ ...
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1answer
166 views

Product of two series to get a series decomposition of zeta in the critical strip

$\def\sfrac#1#2{% \small#1% \kern-.05em\lower0.1ex/\kern-.025em% \lower0.4ex\small#2}$I've been working on gaining an intuitive understanding of the analytic continuation of the zeta ...
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17 views

Uniform and ordinary convergence of a series.

I have a series of the kind $\sum\limits_{n=1}^{\infty} \frac{a_nt^n}{1+t^{2n}}$, $t\in(0,\infty)$. Btw, $a_n$ are real and they are determined by several integrals, but it is possible to compute any ...
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2answers
72 views

Are these two naive upper bounds ok?

EDIT: Is my work ok? Here, I am trying to show a uniform bound for the sum of $cos(n)$ $$|\sum_{n=1}^{N} cos(n)|$$ $$=\big |\sum \frac{e^{in} + e^{-in}}{2}\big|$$ $$\le \sum |\frac{e^{in} + ...
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1answer
50 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer ...
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1answer
42 views

On a generalization for $\sum_{d|n}rad(d)\phi(\frac{n}{d})$ and related questions

Let $\phi(m)$ Euler's totient function and $rad(m)$ the multiplicative function defined by $rad(1)=1$ and, for integers $m>1$ by $rad(m)=\prod_{p\mid m}p$ the product of distinct primes dividing ...
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37 views

Dirichlet series with logs&zeta function

i have the following question: Let F(s) be the Dirichlet series associated to $$f(n)= \sum_{d|n} \frac{log(d)}{d} $$. My answer has to depend on the zeta function. (i.e simplify F(s) so that we can ...
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21 views

Absolute convergence of ordinary Dirichlet series

I am currently reading Serre's 'A course in Arithmetic' and I have a question about proposition 8 of section 2.4 (but I think the question can be answered without knowing the book.) The proposition ...
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49 views

Proof that $\int f(x)\sin(Nx)\ dx \to 0$ as $N \to \infty$

I'm studying Fourier series out of Rudin's "Principals of Mathematical Analysis". In the proof that the Fourier series $s_N(f;x)$ converges pointwise to $f$, it assumes that at a point $x$, there is ...
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48 views

Average Order of $\frac{1}{\mathrm{rad}(n)}$

Again a question about $\mathrm{rad}(n).$ Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, ...
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1answer
62 views

Having some trouble using Dirichlet's test to show series convergence,

The series is $$\sum_{k=1}^{\infty} \frac{1}{k^\alpha}\log\left(1+\frac{1}{k}\right)$$ Since the summand is a product of two factors, and the log factor is monotonically decreasing to zero, as n goes ...
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2answers
66 views

What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
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41 views

Partial GCD - Sum

$\sum\limits_{n = 2}^{M} \sum\limits_{m = 1}^{R} GCD(m,n) $ $R = (\lfloor\dfrac{N}{n}\rfloor-n) \% n $ $M = \lfloor\sqrt{N}\rfloor $ I calculated that - For N=10 the sum is 1 For N = 100 the ...
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1answer
33 views

Multiplicative function and Euler product

Theorem 1 Let $G \subset \mathbb{C}$ be an area and $\sum_{n=1}^{\infty}a_n e^{-\lambda _ns} $ and $\sum_{n=1}^{\infty}b_n e^{-\lambda _ns} $ two Dirichlet-series that converge on $G$ and represents ...
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1answer
114 views

Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?

I've done the following exercise: Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent? My approach: We're going to use the Dirichlet's ...
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61 views

show that there are infinitely many primes congruent to 1 or 4 modulo 5…

Given the following Dirichlet character: $\epsilon (n)=\begin{Bmatrix} 1 : n\equiv 1,4(mod 5)\\ -1 : n\equiv 2,3(mod 5) \end{Bmatrix}$ It is known that it is multiplicative, i.e $\epsilon(nm) = ...
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25 views

Dirichlet kernel [duplicate]

Can anybody help me please in showing that $$\dfrac{1}{2} + \sum_{k = 1}^{N} \cos(kx) = \dfrac{\sin\left(N + \dfrac{1}{2}\right)}{2\sin\dfrac{x}{2}}$$ please?
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1answer
98 views

How does Dirichlet regularization of $1 + 2 + 3 + …$ work?

How does Dirichlet regularization assign value $-1/12$ to $\sum_{k=1}^{\infty} k$? Yes, I know that $\zeta(-1) = - 1/12$, a result that follows from the Riemann functional equation $\zeta(s) = 2^s ...
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1answer
63 views

Dirichlet series expansion for Zeta(s)

Wikipedia lists a series expansion for $\zeta(s)$ here. How is the Dirichlet series below derived? I apologize in advance if this is a very simple question, I don't know much about Dirichlet series. ...
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Integral evaluation for the Riesz means special case $s_n=1$

At the moment, I am investigating the Riesz means defined as the series $$ s^{\delta}(\lambda)=\sum_{n\leq\lambda}\left( 1-\frac{n}{\lambda} \right)^{\delta}s_n. $$ Consider the special case $s_n=1$. ...
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Prove or disprove the existence of f(k) when $\sum_{n=1}^m{\frac{a_n}{n^s}}=\sum_{n=1}^m\frac{a_n+f^n(x)}{n^s}$.

Can we find a function, $f(k)$, such that $$\frac{a_n}{n^s}+\frac{a_{n+1}}{(n+1)^s}+\frac{a_{n+2}}{(n+2)^s}=\frac{a_n+x}{n^s}+\frac{a_{n+1}+f(x)}{(n+1)^s}+\frac{a_{n+2}+f(f(x))}{(n+2)^s}$$ for some ...
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1answer
237 views

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 ...
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1answer
250 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000\ldots}_{n-1\text{ times}} 1 )^T $$ Hence, $$ A_1 =(111111 \ldots )^T $$ ...
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34 views

Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums?

I asked in a previous question whether a function, $a_n$, is unique to $F(s)$ for any Dirichlet function defined by the following $$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$ Its uniqueness property ...
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45 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
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44 views

Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
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1answer
64 views

Does the “alternating” harmonic series where only prime terms are negative converge?

We know that the harmonic series $\sum \frac{1}{n}$ diverges, yet the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges. Euler famously gave a proof of the infinitude (and of the ...
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1answer
151 views

Average order of $\mathrm{rad}(n)$

Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
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1answer
103 views

On the series $\sum_{n=1}^{\infty} (H_{n}+\exp(H_{n})\log(H_{n}))/n^{s}$, where $H_{n}$ is the $n$th harmonic number

It is known the following (see [1], here is an open access PDF on his homepage): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the positive divisors of $n$. Riemann Hypothesis ...
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30 views

What are the coefficients of Modified Fourier Bessel series?

What are the co-efficients of Modified Fourier Bessel series? $$\phi g(r,z,Φ) = \sum_{v=1}^∞ \sum_{m=1}^∞ \sin\left(\frac{mπz}{L}\right) I_v \left(\frac{mπr}{L}\right)\left(A_{vm} \cos(vΦ) + B_{vm} ...
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1answer
94 views

The Dirichlet series $\sum_{n=1}^{\infty}\frac{rad(n)}{n^s}$

Following the example to compute $\zeta (s)\sum_{n=1}^{\infty}\frac{\phi(n)}{n^s}=\zeta (s-1)$, converges absolutely if $\sigma>2$, where $\phi(n)$ is the Euler's totient function and $s=\sigma + ...
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1answer
31 views

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
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1answer
59 views

Sequence Involving Dirichlet Function

The question I have to prove is the following: Let $D(x)$ be Dirichlet Function: $$D(x) = \begin{cases}1 & x\in \Bbb Q \\ 0 & x \notin \Bbb Q \end{cases}$$ Let ...
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1answer
45 views

Prove that these Dirichlet L function are equal to these zeta function products.

Prove or disprove that: $$2 L_{2,1}(s)-2 \zeta (s)+\zeta (s)=\left(1-\frac{1}{2^{s-1}}\right) \zeta (s)$$ $$3 L_{3,1}(s)-3 \zeta (s)+\zeta (s)=\left(1-\frac{1}{3^{s-1}}\right) \zeta (s)$$ Where ...
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1answer
49 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...