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Tagged with zeta-functions riemann-zeta
227 questions
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Riemann Zeta function values at odd & even natural numbers [closed]
If there a general relationship between values of Riemann Zeta function at even natural numbers & odd natural numbers ? Can we get zeta function values of any odd natural numbers using values at ...
1
vote
0
answers
39
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Prove that the function $\bigl(2^{t+c}-1\bigr)\zeta(t)$ is logarithmically concave in $t\in[4,\infty)$
Let
\begin{equation}
c=\log_2\biggl(\frac{1}{16}\frac{60-\pi^2}{15-\pi^2}\biggr)=-0.711\dotsc
\end{equation}
For verifying that the sequence
\begin{equation}
\frac{1}{(2n+1)(n+1)}\frac{2^{2n+2+c}-1}{2^...
- 1,960
8
votes
1
answer
208
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Zeta function related double integral
I recently came across the following double Integral:
$$
I(a)=\int \limits_{0}^{1}\int \limits_{0}^{1}\frac{x^ay^a}{1-xy}dx\,dy
$$
which can be easily shown to be equal to:
$$
I(a)=\zeta(2,a)=\sum \...
- 129
7
votes
0
answers
116
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On the Glaisher numbers and $6^5\sum_{n=0}^\infty\frac1{(6n+1)^5} = \frac{(2^5-1)(3^5-1)}2\,\zeta(5)+\frac{22}{\sqrt3}\,\pi^5$
I. Data
After some experimenting, we find for odd powers $s>1$,
\begin{align}
3^s\sum_{n=0}^\infty\frac1{(3n+1)^s} &= \frac{(3^s-1)}2\,\zeta(s)+\frac{G(s)}{2\, s!\sqrt3}\,(2\pi)^s\\[5pt]
4^s\...
- 56.2k
1
vote
1
answer
52
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A limit involving the Hurwitz Zeta function and binomial coefficients
Let $n \in \mathbb Z_{\geq 3}$. While playing around in Mathematica I noticed the following limit $$B_n(x) \searrow 0 \qquad \text{as }\, x\to\infty,$$
where $B_n(x)$ is an alternating binomial-type ...
- 1,746
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1
answer
44
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Expansion and Simplification of a Hurwitz Zeta Function
On one math page I saw this simplification being made involving the Hurwitz Zeta Function easily simplifying into a normal Zeta Function form: $$\zeta(s,\frac{1}{2})=2^{s}\sum_{k=0}^{\infty}\frac{1}{(...
- 103
1
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0
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40
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is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0 $?
when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞
\frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt $
and ...
1
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1
answer
115
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Finding a closed form for $\sum^{\infty}_{n=1} \frac{1}{(n+1)n^\alpha}$ [duplicate]
I encountered the following sum in my work and I was wondering if it has a known closed form:
$$
\sum^{\infty}_{n=1} \frac{1}{(n+1)n^{\alpha}} \quad , \quad 0 < \alpha < 1 \; , \; \alpha \in \...
- 847
9
votes
3
answers
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"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?
I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
- 258
0
votes
1
answer
89
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Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function
Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
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0
answers
85
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the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $
I was looking at the zero's of
$$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$
for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$.
Now this clearly relates to the Riemann zeta:
$$f(s,a) + \...
- 17.1k
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0
answers
36
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Estimates of the derivatives of $\Xi(s)$
The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$.
This is a problem from my homework: since we can write it ...
- 81
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0
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53
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Zeta Hurwitz function bounds of summation
I have been trying to derive the following equality
$$
\frac{1}{2} \sum_{n=1}^{\infty} \sum_{m=0}^{p-1} \frac{1}{n^s}\left[\cos \left(2 \pi \frac{m q}{p}\left(q^*-1+n\right)\right)+\cos \left(2 \pi \...
- 1
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0
answers
63
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Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta ...
- 45
1
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2
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343
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Proper Way to Calculate Value of Riemann Zeta function?
I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in.
I've been looking at one of the Analytic Continuations of the Zeta ...
- 5,957
1
vote
3
answers
201
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What are some unique integral representations of Apery's constant - $\zeta(3)$?
I've been playing around with some integrals, and I started looking at Apery's constant.
There are some integral representation I've found online, such as:
$$\zeta(3)=\frac{16}{3}\int_{0}^{1}\frac{x\...
- 71
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0
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64
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What are these points of lower magnitude in the Riemann Zeta?
I recently visualized the Riemann Zeta function and noticed an interesting pattern: there are points of slightly lower magnitude along a very slightly curved vertical line, extending furthest out to ...
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2
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1
answer
112
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$|\zeta(1/2 + it)|^2 \geq \frac{\log(t)}{\log \log(t)}$
I've tried to solve this exercise for hours but I didn't managed to figure it out.
Show that there exists a sequence $t \to \infty$ for which
$$|\zeta (1/2 + it)|^2 \geq \frac{\log(t)}{\log\log(t)}$$
...
- 1,406
-1
votes
1
answer
90
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$\sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}$ [duplicate]
Show that
$${\displaystyle \sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}=\sum _{k=0}^{2n}(-1)^{k-1}\,{\frac {\zeta (2k)}{\pi ^{2k}}}\,{\frac {\zeta (4n-2k)}{\pi ^{4n-2k}}}\qquad n\in \...
user1235604
1
vote
1
answer
86
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$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i}\frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)ds$ as a finite sum of $\Lambda(n)$
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 13, question 7]
Express
$$\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)...
- 10.3k
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0
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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 ...
- 10.3k
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0
answers
73
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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
votes
1
answer
95
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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 ...
- 526
-1
votes
1
answer
282
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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
3
votes
1
answer
120
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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 $0$...
- 91
3
votes
0
answers
75
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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 ...
- 3,710
1
vote
0
answers
59
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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 ...
- 39
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1
answer
141
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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 ...
- 101
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0
answers
136
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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 ...
1
vote
0
answers
29
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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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1
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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
11
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3
answers
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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 $\zeta(3)$:
$$\zeta(3)≅\sqrt{\zeta(6)}\left(\frac{1}{\cos \left(\frac{\pi}{18}\right)}+\tan \left(\frac{\pi}{18}\right) \right)=
\frac{\pi^3}{3\...
0
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0
answers
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Integral Representation of the Odd Zeta Function Values
In this book (M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1972), page 807, Equation 23.2.17, it is ...
0
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1
answer
56
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modified riemann zeta function $\zeta ^*(s)$?
I remember there being a function $\zeta ^*(s)$ where
$$\zeta ^*(s)=\zeta (s), \ s\neq 1$$
$$\zeta ^*(1)=\gamma$$
but now I can't seem to find any record of it, does a function like this exist or am I ...
- 369
2
votes
0
answers
61
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Convergence of Riemann Zeta function for curves over the field $\mathbb{F_{q}}$
Let $X$ a curve over the field $\mathbb{F_{q}}$ (proper, integral normal scheme of dimension 1). The Riemann Zeta function for this curve is defined as
$$\zeta(X,t)=\prod_{x\ \text{closed}}\ \left(1-t^...
- 697
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1
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64
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Euler - Mc Laurin summation degree of precision
I'm try to know if the precison of Euler-McLaurin summation depends by the index N inside the formula as follow:
$\zeta(s)_N = \sum_{k=1}^{N} k^{-s} + \frac{N^{1-s}}{1-s} + \int_{N}^{\infty} \frac{x-[...
- 9
1
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0
answers
260
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Meaning of $\zeta(1-s)$ in Riemann Zeta function?
I've been looking at one of the Analytic Continuations of the Zeta function, the Riemann Zeta function:
$$\zeta(s) = 2^s \pi^{s-1} \sin \left(\dfrac{\pi s}2\right) \Gamma(1-s) \zeta(1-s)$$
I ...
- 161
0
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0
answers
246
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Zero-free region for the zeta function
I am studying de la Vallee Poussin's proof of the zero-free region of the Riemann zeta function. Assuming the following bound
$$\Re \frac{\Gamma'}{\Gamma}(s/2) \leq \log \vert s/2\vert + \min \Big(\...
- 39
0
votes
1
answer
108
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an inequality involving the zeta function and its derivative
We know that it can be proved easily that $(s-1)\zeta(s) = s - s(s-1)\int_1^{\infty} \frac{\{x\}}{x^{s+1}}dx$.
I am trying to use this to prove that $\zeta(s) + (s-1)\zeta'(s) > 1 - (2s-1)\int_1^{\...
- 39
1
vote
0
answers
63
views
Is $p^s$ transcendental if $\zeta(s)=0$?
Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers.
Let
$$\zeta_{K}(s)=\prod_{\mathfrak{m}}\frac{1}{1-\#(\mathcal{O}_{K}/\mathfrak{m})^{-s}}$$
be the $\zeta$ function associated to $...
- 1,572
3
votes
0
answers
256
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Explicit Taylor Expansions of Zeta Function
If one's goal were to calculate one value of the zeta function, say $\zeta(-1)$ is at all feasible to do this explicitly by calculating Taylor series and doing an analytic continuation "by hand&...
- 114
0
votes
0
answers
63
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analytic continuation of the spectral zeta function
The spectral zeta function is defined as:
$$\zeta(s)=\sum_{n=1}^\infty \lambda_{n}^{-s}$$
Where $\lambda_n$ is the real spectrum of $L$ (an elliptic self adjoint operator).
I derived an analytic ...
- 204
3
votes
0
answers
48
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Deducing local bound from global bound for zeros of Riemann $\zeta(s)$
I am considering a function related to the Riemann zeta function, and I have a asymptotic for the cardinality of the zeros less than $T$, of the form
$$N(T) = cT^2 + O(T)$$
when $T$ is large. ...
- 567
1
vote
2
answers
83
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Not getting the right answer in $\zeta(0)$
I was checking some results in analytical number theory and came up with two different functional equations for the Riemann's Zeta function.
The first one is
$$\zeta(s) \pi^{-\frac{s}{2}} \Gamma(\frac{...
0
votes
0
answers
45
views
A question on the relation between the complex zeroes of zeta function and the estimate of the error in PNT
I'm currently working on the following problem from Analytic Number Theory.
Assume that $\psi(x)-x=\mathcal{O}(x^a)$, for some $1/2<a<1$, where
$\psi$ is the Chebyshev function.
I would like to ...
user1021241
0
votes
1
answer
100
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Sum of infinite series, Are both of these series equal? 1/2+1/3+1/4...
from an old Numberphile video they explain that the sum of all natural numbers is equal to -1/12, 1+2+3+4+5+...= -1/12. Obviously it diverges, but the -1/12 is meant to be a meaningful representation ...
- 3
0
votes
0
answers
87
views
calculate the integral $\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$
I am trying to calculate the integral $$\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$$
- 237
0
votes
1
answer
147
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The Hurwitz zeta function at the positive integers
Is there a formula that gives the values $\zeta(2n,a)$ as a function of $a$ and Bernoulli numbers, where $n$ is a natural number and $0<a≤1$?
$\zeta(z,a)$ is the Hurwitz zeta function.
- 237
0
votes
0
answers
472
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Contour Integral representation Hurwitz Zeta Function over Hankel Contour
I am trying to prove the following contour integral representation of the Hurwitz zeta Function that appears here.
$$\zeta(s,a)=\frac{\Gamma(1-s)}{2 \pi i}\int_{H}\frac{ z^{s-1}e^{az}}{1-e^z}\,dz \tag{...
- 2,871
0
votes
1
answer
630
views
How to plot Riemann zeta function in xy coordinate system? [closed]
Riemann zeta function in complex plane would look like How to plot this curve best as piece wise functions against x-axis in xy plane.
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