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
66 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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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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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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42 views

A functional equation for a Dirichlet series

I'm looking for a functional equation for the following Dirichlet series $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ?
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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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44 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
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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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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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1answer
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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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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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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
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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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94 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function ($1$ if Rational, $0$ else) can be written as: $$ D(x)=\lim_{m\to\infty}\lim_{n\to\infty} \cos^{2n}(m!\pi x)$$ What is the proof of that? Are those limits ...
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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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2answers
116 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
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1answer
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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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35 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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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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386 views

Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are ...
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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
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
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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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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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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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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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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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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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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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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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
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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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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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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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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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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220 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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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} ...