Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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Conditional convergence of Riemann's $\zeta$'s series

Do Riemann's zeta-function's partial sums $\sum_{n=1}^N n^{-s}$ converge conditionally for some value $s=\sigma+it$ with $\sigma\le 1$? (We must at least have $t\ne 0$ of course.) Partial summation ...
5
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0answers
782 views

Zeta function values in terms of Bernoulli numbers.

The material presented at this link on Zeta function values at even integers proposes a method to compute these that is based on Euler's work. I would like to present a short proof for your ...
3
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1answer
120 views

Identity for $\zeta(k- 1/2) \zeta(2k -1) / \zeta(4k -2)$?

Is there a nice identity known for $$\frac{\zeta(k- \tfrac{1}{2}) \zeta(2k -1)}{\zeta(4k -2)}?$$ (I'm dealing with half-integral $k$.) Equally, an identity for $$\frac{\zeta(s) \zeta(2s)}{\zeta(4s)}$$ ...
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1answer
126 views

Riemann's Zeta function [duplicate]

Possible Duplicate: Riemann Zeta Function and Analytic Continuation Calculating the Zeroes of the Riemann-Zeta function It is stated that Riemann's Zeta function has zeros at negative ...
4
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1answer
152 views

Are these two facts related?

I was told some days ago that the possibility of two randomly picked numbers are relatively prime to each other is $6/(\pi^2)$. And it is well known that the value of Riemann zeta function at 2 is ...
30
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1answer
771 views

A Bernoulli number identity and evaluating $\zeta$ at even integers

Sometime back I made a claim here that the proof for $\zeta(4)$ can be extended to all even numbers. I tried doing this but I face a stumbling block. Let me explain the problem in detail here. I was ...
3
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1answer
213 views

Is it possible to solve this equation analytically?

The following equation holds: \begin{align} & \frac{9}{2}\pi \\[8pt] & = x \\[8pt] & {}+\cos (x) \\[8pt] & {}+\cos (x+\cos (x)) \\[8pt] & {}+\cos (x+\cos (x)+\cos (x+\cos (x))) ...
3
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1answer
394 views

Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
3
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0answers
161 views

How to get the floor function as a Mellin inverse of the Hadamard product of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
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4answers
103 views

Does Zeta converge for Complex numbers Re(s)>1 where imaginary part is not zero?

I know values of zeta for s= 2,3,4,... but what's the value of zeta as an example s= 2+14i
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2answers
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What is the explanation for similar decimal digits in values of Riemann zeta function with certain arguments close to one?

In Mathematica I tried these values close to one as arguments for the Riemann zeta function: ...
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0answers
154 views

If RH is false , could this be true?

Let $\zeta(s)$ be the Riemann zeta function. Assume RH is false , is it possible that we have in the critical strip $\zeta(a_1+ti) = \zeta(a_2+ti) = \zeta(a_3+ti) = \cdots = \zeta(a_n+ti) = 0$ For ...
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1answer
116 views

Hurwitz zeta function

I am using the below function to compute the Hurwitz zeta function from Riemann zeta function. But I am not getting the correct results when compared with the value of Wolfram alpha Hurwitz zeta ...
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vote
1answer
207 views

How does $\zeta(1 - s)$ become $(-1/s + \cdots)$?

Why is $$\zeta(1 - s) = -\frac{1}{s} + \cdots$$ for small negative values of $s$? A detailed explanation would be appreciated.
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Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
2
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2answers
250 views

Sum of the Stieltjes constants? (divergent summation)

The sequence of Stieltjes-constants diverges and thus cannot be summed conventionally. However their signs oscillate (unfortunately non-periodic) and thus I tried Euler- and a version of ...
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0answers
135 views

Approximation of distribution of $\pi_k(n)$ using $\zeta(s)$

Let $\pi_k(n) $ be the number of numbers with k prime factors (repetitions included) less than or equal to n. If we take the sums: $z_1(s) = \sum_{n= 1}^\infty \frac{1}{(p_{1,n})^s},~ z_2(s) = \sum ...
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1answer
708 views

What is the formula for the first Riemann zeta zero?

I found this approximation of which an earlier version I posted in the chat room: $$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 ...
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1answer
64 views

Can this expression be simplified and is the real part of the left hand side equal to minus pi?

By starting with some Dirichlet series similar to logarithms, or somewhat similar to logarithm Dirichlet series, I arrived at this expression: ...
7
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1answer
240 views

Zeta functions of groups and an identity of Ramanujan

Zeta functions are being developed for all sorts of mathematical objects these days. One general situation is that of zeta functions of groups. If $G$ is a finitely generated group then we let ...
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2answers
285 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
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2answers
246 views

Detailed proof of $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$

I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to ...
5
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3answers
790 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
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2answers
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Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
5
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1answer
155 views

Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind

Consider the integral $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$ For $n=3$ we have ...
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0answers
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Riemann Siegel formula modification?

$$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$$ here $N(t)$ is the smooth part of the zeros and $ p_{n} $ are the primes since $ p_{n} =n\log n $ then $ \log ...
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0answers
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fastest way to evaluate $\arg\zeta\left(\frac{1}{2}+i\text{t}\right) $ [duplicate]

Possible Duplicate: evaluation of $ \operatorname{Arg}\zeta (1/2+is) $ ?? If we consider $$\arg\zeta\left(\frac{1}{2} + i\text{t}\right) = \text{Im ...
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0answers
211 views

Are the amplitudes of these frequency spikes equal to 1 when the real part of the complex number “s” is equal to one half?

Over at stack overflow I asked a question about how to plot the Riemann zeta zero spectrum from the von Mangoldt function. Then I asked a question about calculating the Riemann zeta function at the ...
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4answers
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Formula for partial sum of Riemann zeta function [duplicate]

Possible Duplicate: Finite Sum of Power? Suppose $f(s,k) = \sum_{n=1}^k n^{-s}$ is the Riemann zeta function truncated at the k-th term. I read on mathoverflow that there is a formula for ...
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2answers
129 views

Why is the probability that a prime p is a factor of a number n equal to 1/p

I'm learning some number theory and I can't seem to understand why this is the case.
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1answer
160 views

Two Representations of $\log \zeta$

I was looking for representations of $\log \zeta$ and found these two: $ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
7
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1answer
216 views

Improper integral about exp appeared in Titchmarsh's book on the zeta function

May I ask how to do the following integration? $$\int_0^\infty \frac{e^{-(\pi n^{2}/x) -(\pi t^2 x)}}{\sqrt{x}} dx $$ where $t>0$, $n$ a positive integer. This came up on page 32 (image) of ...
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3answers
264 views

Riemann Zeta formula

can anyone check if this formula is plausible ?? $$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$ according to the authors this formula would be valid only ...
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0answers
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Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
5
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2answers
150 views

Riemann zeta sums and harmonic numbers

Given the nth harmonic number of order s, $$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$ It can be empirically observed that, for $s > 2$, then, $$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = ...
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1answer
3k views

Explanation of Zeta function and why 1+2+3+4+… = -1/12 [duplicate]

Possible Duplicate: Why does $1+2+3+\dots = {-1\over 12}$? I found this article on Wikipedia which claims that $\sum\limits_{n=0}^\infty n=-1/12$. Can anyone give a simple and short summary ...
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2answers
467 views

A new formula for Apery's constant and other zeta(s)?

I recently found these Plouffe-like formulas using Mathematica's LatticeReduce. Has anybody seen/can prove these are indeed true? $$\begin{aligned}\frac{3}{2}\,\zeta(3) &= ...
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1answer
329 views

On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, $\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$ It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
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1answer
118 views

Relation between zeta value and genus of modular curve

This question is sort of vague, so I don't mind a vague answer. We have the special value formula $\zeta(-1)=-B_2/2 = -1/12$, where $\zeta$ is the Riemann zeta function. Also, the "genus" of the ...
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1answer
163 views

Simple clarification - deduction using big-O notation

A set of lecture notes I'm reading on Halasz's theorem makes the following statement in a proof, which I can't quite follow - I was hoping someone might be able to clear up what I'm missing: ...
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Is this formula for $\zeta(15)$ true?

Apery gave, $\begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$ J. Borwein and D. Bradley found this can be generalized to ...
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3answers
331 views

Other functional equations for $\zeta(s)$?

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
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1answer
389 views

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
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1answer
630 views

Riemann's thinking on symmetrizing the zeta functional equation

In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as ...
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1answer
180 views

Limit of the function $\zeta(x)/\zeta(x+1)$ as $x \to \infty$

I am looking for a simple proof that $\zeta(\alpha)/\zeta(\alpha+1) \to 1$ as $\alpha \to \infty$ (where $\zeta(\alpha)$ denotes the Riemann zeta function, $\zeta(\alpha) = \sum \limits_{n\geq 1} ...
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1answer
795 views

A new continued fraction for Apéry's constant, $\zeta(3)$?

As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as $\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$ where the ...
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1answer
266 views

is it possible to get the Riemann zeros

since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $ is then possible to get the inverse function $ N(E)^{-1}$ ...
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5answers
2k views

What is so interesting about the zeroes of the $\zeta$ function

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
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4answers
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What do I need to know to understand the Riemann hypothesis

Which kinds of fields of mathematics do I have to know about in order to understand the Riemann hypothesis millenium prize problem?
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1answer
290 views

Does this $\zeta(s)$ identity have a name?

I have generalized the product from this thread: Let $s=2n+1$ for $n\ge1$, $$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...