For questions on Dirichlet series.

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2
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0answers
20 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 ...
1
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2answers
27 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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4answers
78 views

If $\sum x_n$ converges absolutely . I'll have to show that $\sum \frac{x_n }{1+ x_n} $ converges . [duplicate]

I tried applying Dirichlet's test and Abel's test. Here I found some similar questions. But in most of the questions, they have assumed $x_n$ to be greater than 'zero' for all 'n', which allows them ...
5
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1answer
160 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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0answers
16 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 ...
0
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2answers
71 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} + ...
1
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1answer
48 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 ...
3
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1answer
34 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 ...
1
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0answers
34 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 ...
3
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0answers
18 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 ...
0
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0answers
47 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 ...
3
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0answers
42 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, ...
1
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1answer
53 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 ...
2
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2answers
63 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 ...
0
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0answers
35 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 ...
0
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1answer
30 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 ...
6
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1answer
91 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 ...
0
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0answers
52 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) = ...
0
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0answers
21 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?
0
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1answer
88 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 ...
2
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1answer
62 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. ...
3
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0answers
19 views

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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0answers
10 views

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 ...
1
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1answer
233 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 ...
7
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1answer
243 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, $$ A_1 =(111111 \ldots )^T $$ ...
1
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0answers
29 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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0answers
40 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 ...
0
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0answers
38 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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0answers
51 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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0answers
21 views

An exercise on the mulitplication of dirichlet series

This is an exercise problem in the chapter 7 of the Stein complex analysis book. I am stuck at (a). In fact, I have no idea how to deal with the condition "mk = n". Could anyone please help me?
5
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1answer
137 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{ ...
4
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1answer
99 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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0answers
26 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} ...
2
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1answer
81 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 + ...
1
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1answer
30 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 ...
2
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1answer
47 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
43 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 ...
3
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1answer
44 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 ...
2
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0answers
204 views

Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?

The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as: $$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$ It has the following Euler product (I used that Dirichlet character ...
0
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1answer
66 views

How do we prove Dirichlet L-series converges when $\chi$ is non-trivial character and $s>0$?

Let $\chi$ be a non-trivial Dirichlet character modulo $a$. Also assume $a$ has a primitive root $r$. Prove that Dirichlet $L$ function $L(s,\chi)=\sum_{n=1}^\infty {\chi(n)\over n^{s}}$ converges ...
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0answers
23 views

What kind of generating function adapts well both argument shift and multiplicativity?

I have encountered a sequence which involves both some multiplicative arithmetic functions and argument shifts and does not seem to fit neither into Dirichlet nor Lambert kind of generating function. ...
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1answer
36 views

Expression of coefficients of a product of Dirichlet polynomials

Suppose we have two Dirichlet polynomials: $$ f_1(s) = \sum_{n=1}^{m} \frac{a_n}{n^s} \\ f_2(s) = \sum_{n=1}^{m} \frac{b_n}{n^s} $$ Their product will also be a Dirichlet polynomial: $$ ...
3
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2answers
138 views

Showing $\sum\frac{\sin(nx)}{n}$ converges pointwise

I do not understand how one can say using "Dirichlet conditions" that $\sum_{n=1}^{\infty}\dfrac{\sin(nx)}{x}$ is pointwise convergent. I know the proof for $x=1$ but how can one show it is convergent ...
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0answers
28 views

Can you give an example of an irreducible element of the ring of Dirichlet series with integer coefficients?

According to this. The ring of Dirichlet series with integer coefficients is a UFD. Can you give an example of an irreducible element in that ring?
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2answers
66 views

Dirichlet $L$ functions at $s=2$

Let $\chi$ be a Dirichlet character and let $L(\chi,s)$ denotes its Dirichlet $L$-function. What is the value of $L(2,\chi)$ ? Or simply, is $L(2,\chi)/\pi^2$ rational ? Many thanks for your answer ...
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1answer
50 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
2
votes
1answer
47 views

$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest. Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where ...
0
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0answers
22 views

Dirichlet Series of weakly multiplicative characters

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ ...
2
votes
1answer
93 views

Integral of a Dirichlet Series

I'm stuck at a problem of an exercise list... I'd like some help to solve it :) The problem: Suppose that the Dirichlet Series $$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$ has abscissa of ...
4
votes
0answers
60 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...