# Tagged Questions

For questions on Dirichlet series.

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### 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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### 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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### Mobius function related series and Dirichlet summation

A well known identities of rare beauty is that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac 1{\zeta(s)}$$ Where $\mu(n)$ is the Mobius function and $\zeta(s)$ is the Riemann Zeta function. So, in ...
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### Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...
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From previous post* dedicated to aliquot sequences I believe that I can state that for $\Re s>2$, on assumption that the Catalan-Dickson conjecture is false $$\sum_{n=1}^{\infty}\frac{s^{k+1}(n)-\... 0answers 24 views ### Doubts and computations about Dirichlet series and aliquot sequences I Perhaps the more easier statement that one can deduce for aliquot sequences (which is the Wikipedia's Page) is the following Lemma. For an integer n\geq 1, let s^0(n)\equiv n, s(n)\equiv s^1(... 0answers 27 views ### Dirichlet series decomposition of an arbitrary function Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original function at every point. Is there a corresponding notion for Dirichlet ... 0answers 22 views ### 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 ... 0answers 9 views ### 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 ... 0answers 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 ... 0answers 18 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 ... 0answers 50 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 ... 0answers 45 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 ... 0answers 64 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) = \... 0answers 48 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: |x|... 0answers 31 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} \...
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 \$ \textit{gcd}(m,n)...