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Tagged with dirichlet-series riemann-zeta
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"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
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$ 0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?
Define $f(s)$ as
$$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$
where we take the upper complex plane as everywhere analytic.
Notice this is an antiderivative of the Riemann Zeta function, ...
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Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]
I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
user1236730
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How do we increase the region of convergence for the Riemann Zeta function (using Dirichlet Series form)?
The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that:
An ...
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For what values of $c$ is $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0$ when $x=\exp \left(-\frac{\rho _1}{c}\right)$?
The alternating Dirichlet series, the Dirichlet eta function, can be written in the form: $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}$
For what values of $c$ is $$\sum _{k=1}^{\infty } (-1)^{k+1}...
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About the definition of generalized harmonic numbers and an identity
Some software packages make use of the following definition for generalized harmonic numbers. In what follows, $\sigma,t\in\mathbb{R}$:
$$H_{ t }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \
(\sigma+it, t ...
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The evaluation of the coefficient of the Dirichlet series $\zeta'(s)^2$
The derivative of Riemann zeta function is $\zeta'(s)=-\sum_{n=2}^{\infty}(\log{n}) n^{-s}.$
The square of $\zeta'$ is the following Dirichlet series:
$$\zeta'(s)^2=\sum_{n=4}^{\infty}a_nn^{-s},$$
...
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Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
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Convolution Method for Bound
I am reading A survey of gcd-sum functions where the following result is stated:
Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by:
$$\sum_{n=1}^\infty \frac{P(...
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Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$
The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function
$$\eta(s)=\underset{K\to\infty}{\text{lim}...
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Dirichlet series for $\frac{\zeta(1-s)}{\zeta(s)}$ [closed]
Wikipedia (here) says that $\frac{\zeta(s-1)}{\zeta(s)}= \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}$ where $\varphi(n)$ is the totient function. Similarly, is there a known expression involving a ...
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
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How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
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Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$
Let: $$s=\rho _1$$ ...
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