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

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18
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2answers
598 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
0
votes
0answers
32 views

Logarithmic derivative of Riemann zeta, is this derivation correct?

Let matrix $T_2$ be defined below as the Dirichlet inverse of the Euler totient function as a function of the Greatest Common Divisor (GCD) of row index $n$ and column index $k$; $$T_2(n,k) = ...
4
votes
1answer
94 views

Riemann zeta function - Euler product formula

I want to prove that $$ \frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}.$$ I know that the standard proof works with the Euler product formula $$\zeta(s)=\prod_{p \ \text{prime}} ...
4
votes
2answers
192 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
7
votes
1answer
130 views

Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: \begin{equation} \zeta(2n)=c\pi^{2n}, \end{equation} where $n$ is a positive integer ...
-1
votes
1answer
37 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
5
votes
4answers
187 views

Find the regularized sum $\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+…$

By considering the integral Zeta function $$F(s)=s+\frac{1}{2^s\ln(2)}+\frac{1}{3^s\ln(3)}+\frac{1}{4^s\ln(4)}+...$$ Evaluate $$\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+...$$ EDIT: ...
6
votes
0answers
73 views

Why is $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$?

I have a question concerning the Riemann zeta function for a project I've been working on. Why is it that $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$ (that is, there are no non-trivial zeroes of the ...
5
votes
0answers
103 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
2
votes
2answers
78 views

Trivial zeros of the Riemann Zeta function

A question that has been puzzling me for quite some time now: Why is the value of the Riemann Zeta function equal to $0$ for every even negative number? I assume that even negative refers to the ...
1
vote
2answers
80 views

Find the sum $\sum_{n = 1}^{\infty}(-1)^{n + 1}\log(1 + (1/n))$

I started as follows $$\begin{aligned}S &= \sum_{n = 1}^{\infty}(-1)^{n + 1}\log\left(1 + \frac{1}{n}\right)\\ &= \sum_{n = 1}^{\infty}(-1)^{n + 1}\sum_{k = 1}^{\infty}(-1)^{k + 1}\frac{1}{k ...
4
votes
2answers
157 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
6
votes
2answers
288 views

Problem in Divergent series

Lets try to evaluate $$\frac{(-1)}{1^s}+\frac{(-1)^2}{2^s}+\frac{(-1)^3}{3^s}+...$$ $$=\frac{e^{\pi i}}{1^s}+\frac{e^{2\pi i}}{2^s}+\frac{e^{3\pi i}}{3^s}+...$$ $$=\frac{1}{1^s}(1+\frac{\pi ...
5
votes
1answer
122 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
2
votes
1answer
74 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
vote
1answer
90 views

Non-trivial zeros off critical line

If non-trivial zeros lay off the critical line (as shown in the picture below), would they have to come in fours rather than conjugate pairs (as the diagram shows)? I am presuming they would, ...
1
vote
0answers
35 views

Upper bound on $\zeta(s)$

I'd like to know an upper bound for $\zeta(s)$ in the critical strip, and hopefully one that is not too difficult to prove. For instance, ...
0
votes
0answers
30 views

Question about the zeros and poles of the PrimeZeta function.

The Euler product over all primes, $$\displaystyle \zeta(s) := \prod_{p\in\mathbb{P}} \dfrac{1}{1-\dfrac{1}{(p)^s}}$$ is only valid for $\Re(s) >1$. However, when taking the log on both sides ...
1
vote
0answers
72 views

Laurent Series of Riemann Zeta Function

How do I go about finding the Laurent series of the Riemann zeta function about $z=1$?
2
votes
1answer
95 views

Analytic Continuation of Riemann Zeta Function

How do we show that this holds for $\operatorname{Re}(z)>0$ (and not $1$) $$\zeta(z)= \sum_{i=0}^{m-1} n^{-i} + \frac{m^{-z}}2 +\frac{m^{1-z}}{z-1} -z\int_{m}^{\infty} ...
3
votes
1answer
59 views

Logarithms, prove this limit.

Mathematica knows that: $$\log (n)=\lim_{s\to 1} \, \left(1-\frac{1}{n^{s-1}}\right) \zeta (s)$$ Kind of tautological starting with logarithms, but I would like to know better why this limit works: ...
0
votes
1answer
43 views

Sum of zeta(2s) fractions without pi^(2s) in the numerators

$$ \sum _{n=1}^{\infty } \sum _{r=1}^{\infty } (\pi r)^{-2 n}=\frac{1}{2} (1-1 \cot(1)) $$ $\frac{1}{2} (1-1 \cot(1))$ is not in OEIS, so it doesn't seem to be well known. Q1: Would this info be of ...
1
vote
1answer
121 views

Is there a power series representation of $\frac{1}{\zeta(s)}$?

I've seen power series representations for $\zeta(s)$, $\textit{e.g.}$ \begin{align*} \zeta(s)=\frac{1}{s-1}+\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k!}\gamma_{k}(s-1)^{k} \end{align*} where $\gamma_{k}$ ...
3
votes
0answers
57 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
1
vote
1answer
83 views

The alternating zeta function and functional equation

The Dirichlet eta function (the alternating zeta function) is given by $$η(s)=∑_{n=1}^{∞}(-1)ⁿ⁻¹/n^{s}$$ The functional equation for $η(s)$ is given by $$η(s)=ϕ(s)η(1-s)$$ where ...
2
votes
1answer
152 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
0
votes
0answers
57 views

Specific form of integral representation of the Riemann zeta function

Is there an integral represenation of the Riemann zeta function of the form: $$\zeta(s) = f(s)+c\int_a^b\frac{g(x)}{x^{p(s)}}dx,$$ where $a,b,c\in\mathbb{R}$ with $a\neq b$, $p(s)$ is some ...
5
votes
2answers
114 views

$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed ...
0
votes
1answer
31 views

Is there way to write the following integral representations of integer zeta values in a general form?

How do you write this as a general formula for all integer zeta values greater than $0$? $$\zeta(1)=\int_0^1 \frac{1}{1-x}dx$$ $$\zeta(2)=\int_0^1\int_0^1\frac{1}{1-xy}dxdy$$ ...
2
votes
1answer
182 views

Integral representation of the Riemann zeta function

I've come across the following integral representation for the Riemann zeta function, $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi ...
1
vote
1answer
52 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
1
vote
2answers
140 views

Information about Riemann Zeta function

I have general question on Riemann Zeta function. How can I improve knowledge on Riemann Zeta Function theory up to research? For example , what are the best books on Zeta ...
0
votes
0answers
57 views

Differential equation to model a pseudo-random behavior?

In the paper http://arxiv.org/abs/1401.3620, "The zeros of the Riemann zeta-function and the transition from pseudo-random to harmonic behavior", the author built a function based on a finite amount ...
1
vote
0answers
71 views

Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
1
vote
0answers
32 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
5
votes
2answers
98 views

Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
6
votes
3answers
711 views

$1 + 1 + 1 +\cdots = -\frac{1}{2}$

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
1
vote
0answers
58 views

Are the zeros of the sum/difference of two reflexive, entire functions all on the line $\Re(s)=\frac12$?

Remove the order $1$ pole of $\zeta(s)$ at $s=1$, to create the following entire function: $$z(s):=\zeta(s)-\dfrac{1}{s-1}$$ I like to conjecture that all complex zeros of $z(s) \pm z(1-s)$ in the ...
3
votes
1answer
188 views

Calculating $\zeta(0)$ by the residue of $\zeta(1)$

$$\begin {aligned}\pi^{-s/2}\Gamma(s/2)\zeta(s)=&\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \\ \zeta(0) =&\frac{\pi^{-1/2}\Gamma(1/2)\zeta(1)}{\pi^{0}\Gamma(0)}=\frac{\zeta(1)}{\Gamma(0)}\end ...
2
votes
0answers
107 views

A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
2
votes
0answers
91 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
11
votes
2answers
273 views

Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
1
vote
1answer
137 views

Closed form for $\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1)$

From WolframAlpha it seems that $$ \frac{1}{2}=\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1) $$ Could someone provide a proof for this? Thanks.
1
vote
1answer
167 views

Have all the zeros of the Riemann Zeta function real part smaller than 1?

I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$ have real part ...
1
vote
3answers
261 views

Does $\zeta(-1)=-1/12$ or $\zeta(-1) \to -1/12$? [duplicate]

I saw NumberPhile channel on Youtube, and they proved $1+2+3+\cdots=-1/12$. Also, I read This. So, which one is correct $$\zeta(-1)=-1/12\\ \text{or} \\\zeta(-1) \to -1/12$$ Equivalent to: ...
35
votes
3answers
1k views

Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?

Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge? My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here. ...
3
votes
0answers
116 views

Computing Zeta(2)

It is well known that $$\int_0^1\frac{\log{x}}{1 - x}\,\mathrm{d}x = -\frac{\pi^2}{6} $$ This is generally proved by expanding the geometric series and then using $\zeta(2) = \pi^2/6$. My question ...
3
votes
0answers
62 views

Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
0
votes
2answers
136 views

Value of Riemann zeta function at $-1$

This claim is false $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}n^{-(-1)}= \zeta(-1)=-1/12$. The error is that we should $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}(1/n ^1)^{-1}=(0)^{-1}$. Am I correct? ...
1
vote
1answer
191 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...