# Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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### Question on term of derived formula for $\log\zeta(s)$

The derived formula $$\log\zeta(s)=-\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n\in\mathbb{P}} \left(2 \tanh ^{-1}\left(1-2 n^s\right)-i \pi\right)\right),\quad s>1\tag{1}$$ ...
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### Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit $$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$ It is clear that the ...
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As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $... 3 votes 2 answers 71 views ### Upper bound for L-series of modular form from its integral representation I've read that if$f$is a cuspidal modular form (that's also an eigenfunction for the Hecke operators) for$SL_{2}(\mathbb{Z})$of weight$k$, then its L-series$L(w,f)$satisfies the bound $$L(w,f) &... 0 votes 0 answers 27 views ### Dirichlet series of a Gaussian Let a > 0 and s \in \mathbb{C}. I'm wondering whether the series$$ \sum_{n=1}^\infty \dfrac{e^{-an^2}}{n^s} $$are encountered in the literature? Specifically I'm interested in the following: ... 0 votes 1 answer 32 views ### proving that dirichlet series has non negative coefficients and does not converge for all s\in\mathbb{C} given Z(s)=\zeta^2(s)\zeta(s+it)\zeta(s-it) I need to prove that Z(s) is represented by a dirichlet series with non negative coefficients whiche does not converge for all s\in\mathbb{C}. I have ... 5 votes 2 answers 116 views ### Proof that the series \sum_{n=2}^{\infty}\frac{[\Omega(n)]^\alpha}{n^2} converges Let's consider the series$$f(\alpha)=\sum_{n\gt1}\frac{[\,\Omega(n)\,]^\alpha}{n^2}$$where \Omega(n) denotes the number of prime factors of n counted with their multiplicity and \alpha\ge0 is ... 1 vote 1 answer 71 views ### Dirichlet series for \zeta^3(s)/\zeta(2s). I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows: Let ... 1 vote 2 answers 71 views ### Show that \sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}. I am a graduate student of Mathematics. I have started reading number theory. I encountered a problem of analytic number theory. Show that \sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{... 1 vote 1 answer 37 views ### Necessary and sufficient condition to be completely multiplicative I want to prove that f*f=f \tau iff f is completely multiplicative. The "if" part was relatively easy, using f(g*h)=(fg)*(fh) and plug g=h=1 for all n. Juxtaposition is ordinary, ... 0 votes 1 answer 31 views ### Looking for table of special values of the Dirichlet L-function For double checking calculations I made I'd like to find a table of values of L(-1,\chi_D) for small positive fundamental discriminats D. It there a table somewhere in the internet? Where? With \... 1 vote 2 answers 117 views ### Proving \prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)} By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ... 4 votes 1 answer 140 views ### Proof that Dirichlet series \sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2} So I want to prove the following:$$\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2},$$where \omega(n) is the number of distinct prime factors of n. I computed it to 10^{10} and it does ... 1 vote 2 answers 127 views ### Prove \sum_{d | n} \mu(d) (\log(d))^2=0 [duplicate] If n is a positive integer with more than 2 distinct prime factors, how to prove that \sum_{d | n} \mu(d) (\log(d))^2=0? I struggle on how to continue from this. Suppose n=p_1 p_2 ... p_r, ... 1 vote 0 answers 85 views ### Are there are other known functions that give the same set of zeros as the Riemann zeta function inside the critical strip besides \eta(s)? The zeros of the Dirichlet's eta function \eta(s) inside the critical strip match the non-trivial zeros of the Riemann zeta function \zeta(s), as \eta(s) = (1 - 2^{1-s}) \zeta(s). What are the ... 0 votes 1 answer 116 views ### If the Dirichlet's eta function is conditionally convergent in the critical strip how can we ever compute its values there? I'm studying infinite series, but I'm a physicist, not a mathematician. I got it from Hardy's THE GENERAL THEORY OF DIRICHLET'S SERIES that the Dirichlet series \eta(s) = 1^{-s} - 2^{-s} + 3^{-s} - \... 0 votes 0 answers 142 views ### Proof Riemann Zeta Series based on \eta(s) has only one pole. This proof is my understanding of a very interesting comment by @leoli1 on my previous related question about the following extended Riemann Zeta function which converges for \sigma>0 where s=\... 0 votes 0 answers 70 views ### What is the abscissa of convergence of the series \sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ ? What is the abscissa of convergence of the series \sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ ? In the lecture note our instructor claimed that the abscissa of convergence of the above ... 0 votes 0 answers 60 views ### A question about Dirichelet Series. I am looking at some of the formulea here ... https://en.wikipedia.org/wiki/Dirichlet_series Show \begin{eqnarray*} \frac{ \zeta(s) \zeta(s-a) \zeta(s-2a)}{ \zeta(2s-2a) } = \sum_{n=1}^{\infty} \frac{... 1 vote 0 answers 35 views ### Variant of Möbius inversion: b(n) = \sum_{d^2 \mid n} a(n/d^2) d^\alpha I'm trying to understand a step in a classic paper of Rankin. In Rankin's paper Contributions to the theory of Ramanujan's function \tau(n) and similar arithmetical functions, he defines$$ b(n) := \... 4 votes 0 answers 124 views ### What do we know about the analytic continuations of Dirichlet series? Let$s=\sigma+it$be a complex number and define the function: $$F(s)=\sum_{k=2}^{\infty}\frac{p_\pi(k)}{k^s}$$ Where$p_\pi(k)$is the number of unordered factorizations of$k$, corresponding to OEIS ... 1 vote 0 answers 25 views ### Summary of Dirichlet Series Convergence from Apostol's IANT I've been trying to learn about Dirichlet series, in particular from Apostol's IANT textbook. The textbooks tend to present result and not discuss them narratively, so I am left unsure of my correct ... 1 vote 1 answer 46 views ### Step in Apostol's IANT on Dirichlet Series I can't explain the following step in Apostol's IANT regarding Dirichlet Series. Specifically, how does the magnitude of the following $$\left | \int_a^b t^{s_0 -s -1} \right |$$ become the following?... 2 votes 0 answers 64 views ### Question related to potential closed-form representation of Catalan's constant The motivation for this question is to find a closed-form representation for Catalan's constant. Formula (1) below for the Dirichlet beta function$\beta(s)$(which I believe is globally convergent) ... 4 votes 2 answers 224 views ### Prove that the infinite$\sum_{\text{ p prime}}\frac{1}{2^p}$is an irrational number. [duplicate] Prove that the infinite$\sum_{\text{ p prime}}\frac{1}{2^p}$is an irrational number. My progress: Suppose $$\omega = \sum_{\text{ p prime}}\frac{1}{2^p}= \frac{1}{2^2}+\frac{1}{2^3}+\dots$$ Also ... 0 votes 0 answers 40 views ### Question on Difference Root representations of$\eta(2 n+1)$and$\beta(2n)$where$n\in\mathbb{N}$I've noticed the Dirichlet eta function$\eta(s)$and the Dirichlet beta function$\beta(s)$can be represented by difference roots at odd and even positive integers respectfully. For example, ... 3 votes 0 answers 95 views ### The average order of the divisor functions${\sigma _\alpha }(n)$, where$\alpha < 0$(Apostol, Intro to Analytic Number Theory, p.61) In Apostol’s book, Theorem 3.6 (p.61) states a result concerning the average order of${\sigma _\alpha }(n)$, where$\alpha < 0$I am including an outline of Apostol’s approach, I hope I have ... 3 votes 1 answer 159 views ### Dirichlet transform of$e^{(2 \pi i / 3) \Omega(n)}$The Dirichlet transform of the Liouville function$\lambda(n)$is famously $$\sum_{n=1} \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\tag{1}$$ The Liouville function is defined by $$\lambda(n) ... 0 votes 2 answers 49 views ### How to find the coefficent of a term in a Dirichlet generating function in Mathematica? For a normal Dirichlet generating function like Zeta[s]^2, I can get the coefficient of the n-th term by applying Dirichlet convolution of the two constant functions. But how to find the coefficient ... 0 votes 1 answer 171 views ### For which s does \sum 1/p^s converge? A well-known result is that \sum 1/n^s converges for \operatorname{Re}(s)>1. Question: For which s does \sum 1/p^s converge, where p is over all primes? Notes: Intuitively there are ... 2 votes 1 answer 52 views ### Product over the primes with relation to the Dirichlet series What is the value of \displaystyle \prod_p\left(1+\frac{p^s}{(p^s-1)^2}\right) I got this product by defining a function a(n) such that a(n)=a(p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_n^{a_n})=a_1a_2a_3...... 2 votes 1 answer 79 views ### Confused by boundedness and convergence of Dirichlet Series (Apostol 11.6 Lemma 2, Theorem 11.8) Apostol's IANT Section 11.6 is on "The half-plane of convergence of a Dirichlet Series". In it he proves that if a Dirichlet series is bounded at s_0 then it is also bounded at \sigma>... 1 vote 0 answers 149 views ### Convergence of a Dirichlet series For a fixed positive integer j, consider the arithmetical function :$$\vartheta _{j}(k+1)=\left\{\begin{matrix} 1 \;\;, & k+1=j^{l}\;\;(l=1,2,3...)\\ 0 \;\;, & \text{otherwise} \end{... 0 votes 1 answer 48 views ### Last step in Apostol's Section 11.6 Lemma 2 (Dirichlet Series) Apostol's IANT Section 11.6 on the half-plane of convergence presents and proves a Lemma 2. Question: I can't understand the simplifications he does at the last step. The image below highlights the ... 1 vote 1 answer 72 views ### Validity of proof showing difference in abscissa of convergence and absolute convergence of Dirichlet Series is$\leq1$? The following is a step-by-step proof/derivation showing the difference in abscissae of convergence$a_c$and absolute convergence$a_a$is never more then 1. Question: Is this simple proof correct? ... 0 votes 0 answers 49 views ### Why is$\lim_{x\rightarrow \infty} \sum_{x<n\leq\infty}a_{n}n^{-s}= 0$a sufficient condition for convergence? Assume the following is true $$\left|\sum_{x_{1}<n\leq x_{2}}\frac{a_{n}}{n^{s}}\right| \leq Kx_{1}^{-\sigma}$$ where$s=\sigma+it$and$a_n$are complex, and all other variables are real, and$\...
Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below) $$\sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt$$ to prove a result about complex ...
This is from Stein and Shakarchi's Complex Analysis, Chapter 7, Exercise 2. Show that if $\{ a_m \}$ and $\{ b_k \}$ are two sequences of complex numbers with bounded partial sums, and $\Re(s) > 0$,...