Questions tagged [zeta-functions]
Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.
762
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$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i}\frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)ds$ as a finite sum of $\Lambda(n)$
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 13, question 7]
Express
$$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i} \frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)...
- 8,425
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For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)]
Let $f(n)$ be an arithmetical function which is periodic mod $k$. If
$$\sum_{r=1}^k f(r)=0$$
then prove that the ...
- 8,425
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0
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23
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What does $(Z - \delta)^k [x,y]$ count?
I am trying to understand my professor's lecture notes about zeta function and using it to count.
DEFINITIONS
The zeta function is defined as $Z[x,y] = \begin{cases} 1 & \text{if
> } x \leq ...
- 309
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44
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"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
2
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1
answer
76
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Proofs involving manipulation of divergent series
Is this proof valid even though the harmonic series it is based on is divergent?
Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$
Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
- 15.3k
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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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A certain formal power series equals the zeta function of a curve
Be $X$ a projective, geometrically connected and smooth curve over the finite field $\mathbb{F}_q$. Then one can define the following series, i.e. the zeta function of the curve: $$Z(t) = \exp\left(\...
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An integral similar to Abel Plana: $\displaystyle \int_{0}^{\infty}{\dfrac{\ln(1+x^2)}{e^{2πnx}-1}}\ \mathrm{d}x\\ $
Recently I've come across three infinite integrals of very similar forms:
$\displaystyle \int_{0}^{\infty}{\dfrac{\ln(1+x^2)}{e^{2πnx}-1}}\ \mathrm{d}x\\ $
$\displaystyle\int_{0}^{\infty}{\dfrac{\ln(1+...
- 151
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If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges. What's wrong in my proof?
If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges.
Clearly this is false in general and only holds for $n=1$ . But here's my wrong proof in which I can't find what's wrong.
Let $$...
- 2,459
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Zeta functions of hyperelliptic curves
I have been wondering recently about the geometric information encoded in the zeta function of a (smooth, projective) variety over a finite field - or in its étale cohomology (i.e. l-adic cohomology) ...
- 63
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2
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119
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Analytic continuation of Dirichlet beta funcion
In terms of Hurwitz zeta function, Dirichlet beta function is given by
$$\beta(s)=\frac1{4^s}\left(\zeta(s,\frac14)+\zeta(s,\frac34)\right).$$
Following the links, by means of analytic continuation of ...
- 8,584
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60
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Cancelling terms in an infinite series over $\mathbb{N}^2$
I try to understand Don Zagiers seemingly simple proof of the recurrence relation
\begin{align*}
\sum_{0<j<k,\ j\ \text{even}}\zeta(j)\zeta(k-j)=\frac{k+1}{2}\zeta(k)\ \ \ \ \ \ \ \ \ \ \ \ \ \...
- 11
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Zeros of Riemann's $\xi(t)$
In Riemann's paper he defined $\xi(t)=\Pi(\frac{s}{2})(s-1)\pi^{-\frac{s}{2}}\zeta(s)$, where $s=\frac{1}{2}+ti$. On page 4 he said:
The number of roots of $\xi(t)=0$, whose real parts lie between $0$...
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Is the function $F(s)= \int_{\gamma_{\epsilon}} \frac{z^{s-1}}{e^{z}-1} \ dz$ analytic?
Let $s \in \mathbb{C}$ . $$\gamma_{\epsilon}= \begin{align*}
\left\{ z \in \mathbb{R}: z : + \infty \to \epsilon\right\} \cup \left\{ \epsilon e^{i\theta} : \theta \in [0,2\pi] \right\} \cup \left\{...
- 1,727
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Function $\mathcal{Z}\left(x\right)=\sum_{n=1}^{\infty}n^{-x}\sin\left(n\right)$ [duplicate]
This is just a modification of the Zeta Function, if it is already present in literature, please link me to it.
$$\mathcal{Z}\left(x\right)=\sum_{n=1}^{\infty}n^{-x}\sin\left(n\right)$$
This is the ...
- 2,663
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Converting polylogarithms to Dirichlet L functions
When trying to simplify polylogarithms evaluated at some root of unity, namely $\text{Li}_s(\omega)$ for $\omega=e^{2\pi i ~r/n}$, it is reasonable to convert it to Hurwitz zeta functions or Dirichlet ...
- 1,529
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31
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Precomposing local L functions with $p^{-x}$
Let $X_0$ be a variety over an algebraically closed field K. In defining $Z(X_0,t) = \prod \frac{1}{1-t^{\deg(y)}}$, we slip in a $t = q^{-x}$ when defining the global $\zeta$-function, to get $\zeta(...
- 5,627
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Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
- 56
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Closed form for $\sum_{n=1}^\infty\binom{2n}{n}\zeta(m,n+1)x^n$
I need a closed form for the following series which appeared when evaluating another sum.
$$S(m,x)=\sum_{n=1}^\infty\binom{2n}{n}\zeta(m,n+1)x^n, \quad m\in\mathbb{Z}$$
I've tried expanding the ...
- 2,107
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1
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What is the relation between Zeta Function and nth Integral?
I was reading the article in Wikipedia about Apery Constant and I saw the triple integral
$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xyz}dx~dy~dz$$
By curiosity I removed the z variable and the ...
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279
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Generalise logarithmic integral $ \int_0^1\ln^n (\frac{1-x}{1+x})\mathrm dx$ related to Zeta function
Context:
I've been recently putting integrals of some random functions in online calculators for fun. An interesting kind of logarithmic integrals that I observed are as follows :
\begin{align}
&\...
- 2,459
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Prove $\frac{ζ\left(\frac{1}{2}+it\right)}{ζ\left(\frac{1}{2}-it\right)}=0$ $⟹\ t=i\left(n-\frac{1}{2}\right),\ n\ ∈\ Z$
Do I prove it works?
$$\frac{ζ\left(x\right)}{ζ\left(1-x\right)}=\frac{2^{\left(x-1\right)}\pi^{x}\csc\left(\frac{1}{2}\left(\pi\left(1-x\right)\right)\right)}{Γ(x)}=\frac{2^{\left(x-1\right)}\pi^{x}\...
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A question from Titchmarsh's book " The Theory of the Riemann zeta function, Theorem 9.16, page 231
I am studying about upper bounds for $N(\sigma, T)$ (zero-density estimates), and while going through Theorem 9.16 in Titchmarsh's book (2nd edition) (page 231), I got a bit stuck in understanding the ...
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can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
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Finding the function that corresponds to a Fourier series
Im trying to reverse engineer a Fourier series that came about in my work by happenstance. The series in question:
$$
\sum^{\infty}_{n=0} \frac{c}{c^{2}+(2n+1)^{2}}\sin((2n+1)x)-\frac{2n+1}{c^{2}+(2n+...
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Relation of Class numbers and Special Values of zeta functions
I am reading Serre's article on p-adic Modular forms (Formes modulaires et fonctions zˆeta p-adiques).
In section 1.6, Serre constructs p-adic Eisenstein series by considering
$G_{k_i} = \frac{1}{2}\...
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A few questions about Riemann's Main Formula in the paper On the Number of Primes Less Than a Given Magnitude
Sorry for asking multiple questions these days about the same topic, but the thing is I was doing a school project about Riemann's zeta function so I kind of suffered when reading Riemann's paper On ...
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Complex properties of the Hurwitz zeta function
Is there any known expression for the complex or real part of the Hurwitz zeta function $\zeta(s,a)$?
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How To Use Complex Contour Integration To Obtain Zeros Of Riemann Zeta Function
Specifically, can someone recommend resources that continue from the end of this YouTube video: https://www.youtube.com/watch?v=uKqC5uHjE4g&t=2s&ab_channel=zetamath?
Additionally, are there ...
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Riemann hypothesis like conjecture for a non-UFD?
I got inspired by quadratic rings and zeta functions. I know for instance that the norm $a^2 + 17 b^2$ for the ring $\Bbb Z[\sqrt{-17}]$ is multiplicative yet the ring $\Bbb Z[\sqrt{-17}]$ is not a ...
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Looking for a specific zeta function.
I am looking for a zeta function
$$ f(s) = \sum \frac{1}{a_n^s}$$
Where $a_n$ is a sequence of distinct positive integers,
such that
$f(s)$ is analytic for all $Re(s) > 1$
$f(s)$ has a simple ...
- 15.3k
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Closed form for $A=\lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k} \frac{1}{a^2 + 17 b^2} - \sum_{k<a^2 + 17 b^2 < k^2} \frac{1}{a^2 + 17 b^2}$?
Does the following double sum converge and if so, do we have a closed form ?
$$ A = \zeta_{17}(1)= \lim_{s=1} \lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k}^{++} \frac{1}{(a^2 + 17 b^2)^s} - \...
- 15.3k
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Zero-free region of $\zeta^{(k)}(s)$
I am new to the Riemann zeta function.
In the beginning of section 2 of Bruce c. Berndt, the number of zeros for $\zeta^{(k)}(s)$, J. London Math. Soc. (2), Vol 2, 1970, p. 577-580, the author made ...
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Basic Riemann-zeta question (summability)
This is a quick question about when the sum of a product is the product of a sum. We have the definition of the zeta function as
\begin{equation}
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.
\end{...
user1022331
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Asymptotic behavior of $\psi^{(n)}(1/2+ix)$ as $n\to\infty$?
I'm trying to determine the behavior of an asymptotic series whose $n$th term (where $n$ is even, and all odd terms vanish) is proportional to
$$\psi^{(n)}\left(\frac12+ix\right),$$
where $\psi^{(n)}$ ...
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Finding residues of a sum in paper by Hitchen
I am reading Nigel Hitchin's paper on Harmonic spinors [Adv. Math. 14 (1974) 1-55]. On p 34 he defines a function
$$
f(s)=
\sum_{p,q\in{\mathbb Z}_+ }(p+q)(4pq \lambda^2 +(p-q)^2)^{-s}.
$$
The ...
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Is this reasonable closed form approximation for the value of $\zeta(3)$ using trigonometric functions just a coincidence?
I have found this expression for an approximate value of $\zeta(3)$:
$$\zeta(3)≅\sqrt{\zeta(6)}\left(\frac{1}{\cos \left(\frac{\pi}{18}\right)}+\tan \left(\frac{\pi}{18}\right) \right)=
\frac{\pi^3}{3\...
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Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?
An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
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evaluate $\int_0^{\infty} e^{-x} \prod_{m=0}^{\infty} ( 1-e^{-2^m x}) \space dx$ to find a fair sharing constant
$$\int_0^{\infty} e^{-x} \prod_{m=0}^{\infty} ( 1-e^{-2^m x}) \space dx$$
Is the amount more than Bob that Alice gets when dividing objects according to the Thue-Morse fair sharing sequence (ABBABAAB.....
- 4,074
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Recover coefficients of convex combination of Hurwitz zeta function
I have coefficients $a_i\in\mathbb{R}$ for $i=1,...,p$ and a term
$$\mathcal{Z}(z)=p^{-z}\sum_{j=1}^p a_j \zeta(z,jp^{-1}),\qquad \sum_{j=1}^p a_j = 1$$
and I was wondering whether I can recover the ...
- 1,071
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47
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Definition of $a=0$ limit of Hurwitz Zeta function
The definition of Hurwitz zeta function is
$$
\zeta(s,a) =\sum_{k=0}^\infty \frac{1}{(k+a)^s}
$$
where $a=0$ limit is obvious singular.
But in functionsite: https://functions.wolfram.com/...
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0
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97
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The explicit expression of a variant of derivative of logarithm of Riemann Zeta function
I find some explicit expression of derivative of logarithm of Riemann Zeta function $\frac{\zeta'(s)}{\zeta(s)}$ is already found. For example, in this answer link.
I am curious about whether there is ...
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0
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Integral Representation of the Odd Zeta Function Values
In this book (M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1972), page 807, Equation 23.2.17, it is ...
1
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0
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38
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Simple and interesting inequality involving the Lerch transcendent function
I’m trying to prove this general inequality related to the Lerch transcendent function:
$$
\ell \cdot \left(\Phi(\frac{1}{\ell + \frac{c}{\sqrt{c^2+1}}}, \frac{1}{2}, c^2) - \frac{1}{c}\right) \geq \...
2
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0
answers
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Numerically evaluating $\zeta(s,a)$ for $s, a \in \mathbb{C}$ given that $\Re(s)\le 1, \Re(a)>0$.
I am currently working on some numerical code to evaluate the Hurwitz zeta function of complex arguments. My approach starts by using DLMF §25.11.4 and §25.11.3 to ensure that $\Re(a) > 0$:
$$
\...
- 131
3
votes
2
answers
246
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For $n\ge m\ge 1$, how far can we walk with $ \int_0^{\frac{\pi}{2}} \frac{x^n}{\sin^m x} d x$?
In the post, I tackled the integral by power series and integration by parts and obtained that
$$
\int_0^{\frac{\pi}{2}} \frac{x^2}{\sin x} d x=2\pi G-\frac{7}{2}\zeta(3)
$$
where $G$ is the Catalan’s ...
- 18k
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1
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Numerator of zeta $Z(C/\Bbb{F}_q, T)$ and Cayley-Hamilton theorem and Frobenius
Let $C$ be a genus $g$ curve. I know numerator of congruent zeta function $Z(C/\Bbb{F}_q, T)=\sum \frac{\sharp C(\Bbb{F}_{q^n})x^n}{n}$ is product of characteristic polynomial of Frobenius, $det(1-...
- 5,555
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Roots and analytic continuation of $\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $?
Let
$$\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $$
Where $H_n$ are the harmonic numbers.
This is well defined for $\Re (s)>1$.
But what about analytic continuation?
And where is
$$\zeta(s) = 0$$
??
...
- 15.3k
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1
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modified riemann zeta function $\zeta ^*(s)$?
I remember there being a function $\zeta ^*(s)$ where
$$\zeta ^*(s)=\zeta (s), \ s\neq 1$$
$$\zeta ^*(1)=\gamma$$
but now I can't seem to find any record of it, does a function like this exist or am I ...
- 319