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

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2
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1answer
56 views

Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
3
votes
2answers
135 views

Real zeros of the zeta function

How does one show that the negative even integers make up all the real zeros of the zeta function? That is, how does one show that there are no real zeros on the interval [0,1]? I am aware that you ...
1
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0answers
50 views

Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip
0
votes
0answers
62 views

Invalid use of the analytic continuation of the Riemann zeta function?

Watching this video on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my ...
3
votes
1answer
51 views

Does this relative primes formula violate the feasibility of picking truly random numbers?

I read a question on this site recently that fascinated me by pointing out that you can't truly pick a random number from an infinite set. I can't find the answer now, but it was shown that you have ...
2
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0answers
35 views

Different methods of calculating $\zeta(s)$'s Laurent series.

Initially, I thought that calculating$$\int_\gamma \frac{\zeta(z)}{(z-1)^n}dz$$ directly, where $n \in \mathbb{Z}$ and $\gamma$ is an anticlockwise contour around $z=1$ with winding number $1$, would ...
2
votes
3answers
116 views

$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
14
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1answer
877 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
2
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0answers
94 views

Could Fibonacci numbers be related to Riemann zeros? [closed]

this is my question can tghe fibonacci numbers $$ F_{n+2} =F_{n+1} +F_{n} $$ be related to the zeros of the Riemann zeta function ?? i heard that in the webpage ...
2
votes
2answers
203 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
2
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1answer
122 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
5
votes
2answers
310 views

A conjecture relating Multiple Zeta Values and the Polya Enumeration Theorem

Let me state my motivation. I believe that the Polya Enumeration Theorem and Multiple Zeta Values (the classic being the Basel problem and the values of the Riemann zeta function at the even ...
1
vote
1answer
89 views

About Riemann's zeta function

Is the riemann zeta function analytic? If so can it be expressed as a power series? Does it have a ratio of convergence ? Could it be said to have a center point of its ratio of convergence at ...
0
votes
1answer
89 views

Riemann Zeta circularity?

In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} ...
3
votes
1answer
99 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
3
votes
0answers
63 views

The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
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1answer
106 views

Contour approach to Riemann zeta functional equation

I have a question regarding Riemann's first proof of the functional equation that was given in his paper on the Riemann zeta function. I am an undergraduate working on a fairly short undergraduate ...
1
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1answer
93 views

How to prove $\sum_{n=1}^{\infty}\frac{\mu (n)}{n^{s}}=\frac{1}{\zeta (s)}$?

How can we prove this equation? $$\sum_{n=1}^{\infty}\frac{\mu (n)}{n^{s}}=\frac{1}{\zeta (s)}$$
21
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2answers
709 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 ...
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0answers
72 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
242 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}} ...
5
votes
3answers
771 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
169 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
41 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 ...
8
votes
4answers
291 views

The sum $\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+…$ is divergent. Find the regularized evaluation

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: ...
8
votes
2answers
121 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
136 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 ...
3
votes
2answers
477 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 ...
2
votes
2answers
103 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
3answers
222 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 ...
7
votes
3answers
327 views

Problem in Divergent series

Lets try to evaluate $$\frac{(-1)}{1^s}+\frac{(-1)^2}{2^s}+\frac{(-1)^3}{3^s}+\cdots$$ $$=\frac{e^{\pi i}}{1^s}+\frac{e^{2\pi i}}{2^s}+\frac{e^{3\pi i}}{3^s}+\cdots$$ $$=\frac{1}{1^s}(1+\frac{\pi ...
6
votes
1answer
180 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 ...
3
votes
1answer
139 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 ...
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1answer
170 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, ...
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0answers
82 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, ...
2
votes
1answer
145 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
71 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
50 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 ...
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1answer
145 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 $\gamma_{k}$ ...
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0answers
79 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 ...
2
votes
1answer
143 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
195 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 ...
6
votes
2answers
154 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
38 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 $0$? $$\zeta(1)=\int_0^1 \frac{1}{1-x}dx$$ $$\zeta(2)=\int_0^1\int_0^1\frac{1}{1-xy}dxdy$$ ...
3
votes
1answer
532 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
108 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 ...
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2answers
260 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 ...
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0answers
90 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 ...
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0answers
50 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
107 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 ...