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Results tagged with riemann-zeta-function
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The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
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Zeros of Dirichlet function $L(s,\chi_4)$
I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function
$$
L_4^* (s,\chi_4) = …
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Scaled Riemann zeta function with no zero in the critical strip
Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues.
Prime numbers are denoted as …
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0
answers
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Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. O …
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Analytic continuation and convergence of a Riemann zeta related function
The functions in question are
$$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\ …
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0
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Abscissa of convergence of transformed Dirichlet series
Let
$$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$
where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a comple …
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0
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1
answer
373
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On some property of the zeros of $\zeta(s)$ in the complex plane
This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well kno …
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Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\s...
Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T
=\Omega_T(\sigma)$ as the maximum ci …
- 3,098
0
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0
answers
101
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Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n …
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About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$
The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$.
$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\log n) …
- 3,098
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0
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Incredibly accurate recursions for the Riemann Zeta function
Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here.
During some statistic …
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From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\log k …
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Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula
I was trying to get some interesting result for $\zeta(3)$, exploring the following function:
$$W(a) = \sum_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim_{a\rightarrow 0} W(a) = \zeta(3).$$
Let …
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2
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1
answer
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Truncated Euler products, Dirichlet eta function, and convergence issues
Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,t)=\sum_ …
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650
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Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of converg …
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Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L_4^*(s) = \p …
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