For questions on Dirichlet series.

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Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1 times} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 = ...
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Derivation of Perron's formula

I tried to derive Perron's formula, but got really screwed up. I would appreciate some pointers on where I'm going wrong here. The problem I'm working on actually requires me to define this for ...
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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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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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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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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?
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202 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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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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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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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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Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} ...
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261 views

Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
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1answer
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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
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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
51 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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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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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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152 views

How write Dirichlet character sums for the terms of the von Mangoldt function?

The way to separately write the terms of the von Mangoldt function $\Lambda$ as Dirichlet character sums seems to be: $$\Lambda (1) = \sum\limits_{n=1}^{\infty } \frac{(e^{\Lambda (1)} \chi ...
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1answer
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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 ...
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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 ...
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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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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: $$ ...
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Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
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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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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
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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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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 ...
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Sines and cosines of angles in arithmetic progression [closed]

Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ ...
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38 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
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1answer
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$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ 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 ...
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Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
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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 $ ...
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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 ...
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1answer
117 views

Prove that this sequence of integers is on average equal to zero.

Consider the sequence $\{a(n)\}_{n\in\mathbb{N}^*}$ that is defined by the Dirichlet series: $$\zeta (s)^2\cdot\left(1-\frac{1}{2^{s-1}}-\frac{1}{3^{s-1}}+\frac{1}{6^{s-1}}\right)=\sum_{n\geq ...
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How to prove $(\frac{1}{5^3}-\frac{1}{7^3})+(\frac{1}{11^3}-\frac{1}{13^3})+(\frac{1}{17^3}-\frac{1}{19^3})+…=(1-\frac{\pi ^3}{18\sqrt{3}})$

How to prove $$ \sum_{k=1}^\infty \left[\frac{1}{(6k-1)^3} - \frac{1}{(6k+1)^3}\right] = 1 - \frac{\pi^3}{18\sqrt{3}}$$ I think this equality likes the Dirichlet Beta function. The numerical value ...
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Which one will give one

We know that the odd terms of Dirichelt Beta function are $\frac{\pi }{4}$,$\frac{\pi^3 }{32}$,$\frac{5\pi^5 }{1536}$,... If we use the WolframAlpha to find the limit of the odd terms only divided ...
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1answer
165 views

Abscissa of convergence for a Dirichlet series

Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = ...
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Special case of prime number theorem for arithmetic progressions 4k+1

In terms of the proof of prime number theorem for arithmetic progressions, I have seen many proofs involving with the concept of "character". Is there an alternative way (without such a concept) to ...
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Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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Calculating the abscissa of convergence for general Dirichlet Series

I'm currently interested in proving this theorem which I have been thinking for quite a while: Define a Dirichlet Series $$\sum_{k=1}^{\infty}a_k e^{-\lambda_k z}$$ where $\lambda_k$ is a strictly ...
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1answer
54 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
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Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
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Can this matrix be said to consist of Dirichlet characters?

When: $j=1$ the following formula: $$T(n,k)=\prod\limits_{m=1}_{m \mid n}^{\text{n}} \left(\exp ^{-\mu \left(\frac{n}{m}\right)}\left(\Lambda \left(\frac{n}{m}\right)\right) \chi _{\exp ^{-\mu ...
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1answer
94 views

Uniform convergence

How would you show that $$f(x)=\sum_{n=2}^\infty \frac{\sin(2\pi n x)}{n\log n}$$ converges uniformly on $x\in[0,1]$. The pointwise convergence can be proved by Drichlet test
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Plotting Dirichlet Series $L_{\tau}(s)$ related to Ramanujan tau function in Mathematica

According to wikipedia, Ramanujan tau function $\tau:\mathbb{N}\to\mathbb{Z}$ is defined by equating the coefficients of the power series on both sides of the identity: ...
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58 views

Why is $\mu \star E =e $ , where $\star$ denotes the Dirichlet Convolution operator?

Let $$ E(n) = 1 \qquad \forall n \in \mathbb{Z} $$ be the constant function, and let $\mu$ be the Möbius function. Based on the following definition of the latter function, where $\mu(n) = 1$ for ...
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1answer
43 views

A converse theorem for Hecke modular forms?

Let $f$ be a function which can be represented by a Dirichlet serie (for $\Re s$ big enough) $$f(s)=\sum_{n=1}^{\infty}{\frac{a(n)}{n^s}}$$ and which can be meromorphically extend to $\mathbb{C}$ by ...
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8 views

The Dirichlet G.F of $2^\Omega$

Let $\Omega(n)$ and $\omega(n)$ be the number of prime factors of $n$ and of distinct prime factors of $n$, respectively. The Dirichlet G.F of $2^\omega$ is well known, and I was wondering if that of ...
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82 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 ...