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

learn more… | top users | synonyms

2
votes
2answers
327 views

I think I found a flaw in Riemann Zeta Function Regularization

I think I may have found a flaw in how Zeta Regularization works. As we all know, it's very famous for proving that $1+2+3+4+...=(-1/12).$ See here (5 rows of equations at the end of this post) ...
0
votes
1answer
102 views

Does Riemann Hypothesis imply strong Goldbach Conjecture? [duplicate]

In Andrew Granville's 2007 paper: "REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS" He said: "an averaged strong form of Goldbach's conjecture is equivalent to the ...
0
votes
0answers
27 views

Selberg's theorem, rotational invariance and circle on the Riemann sphere

If I'm not mistaken, Selberg proved that $\vert\zeta(1/2+it)\vert$ is normally distributed. But the normal distribution is known for its rotational invariance property and as a matter of fact, RH is ...
3
votes
1answer
51 views

Question about complex analysis in proof in Ingham

This is a detail from a proof in Ingham's Distribution of Prime Numbers, p. 91-92. He forms a Dirichlet integral and assumes for contradiction that the numerator $c(x)\geq 0.$ Then he bounds $f(s)$ in ...
0
votes
1answer
98 views

How does Dirichlet regularization of $1 + 2 + 3 + …$ work?

How does Dirichlet regularization assign value $-1/12$ to $\sum_{k=1}^{\infty} k$? Yes, I know that $\zeta(-1) = - 1/12$, a result that follows from the Riemann functional equation $\zeta(s) = 2^s ...
0
votes
1answer
50 views

is there a Globally convergent series for Riemann Xi function?

According to Wikipedia, there is a global convergent series for Riemann Zeta function: https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series Is there a similar global ...
5
votes
0answers
83 views

Can this approximate closed form of Apery's constant $\zeta(3)$ be improved?

I know that an approximate closed form is not really a solution. However, I would like to present a method that gives a closed form of $\zeta(3)$ that is accurate to the 5th decimal, hoping that it ...
0
votes
0answers
63 views

An understandable explanation of the graph of zeta spiral $z(t)=\zeta\left(\frac12+it\right)$

Just as for a graphical real variable, domain and range are studied, their roots, their growth and minimum and maximum, convexity and role that has derivative in previous computations, too its ...
0
votes
0answers
29 views

Clarify mertens' theorem?

If merten's theorem states this http://mathworld.wolfram.com/MertensTheorem.html (equation on the second line) specifically, then what is the error described as for finite n?
3
votes
0answers
66 views

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line?

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line ? Are there particular mechanism to make this happen ? Because of Levinson and Corney's work, we ...
2
votes
1answer
92 views

The values of the derivative of the Riemann zeta function at negative odd integers

I would like to know if the values of the derivative of the Riemann zeta function at negative odd integers are computed, i.e. $\zeta'(-n)$ when $n$ is odd. When I look at the page from Wolfram ...
6
votes
2answers
118 views

Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem?

Here is a question about Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem $?$ In other words, (just for some brain ...
-3
votes
1answer
58 views

Pick the $\zeta(3)$ contribution from Gamma function countour integral

I edited the post and title. How do we see that given $$ Z= \oint \frac{d \epsilon}{2\pi i} (z\bar z)^{-\epsilon} \frac{\pi^4 \sin 5\pi \epsilon}{\sin^5 \pi \epsilon} \left|\sum_{k=0}^\infty (-z)^k ...
3
votes
1answer
84 views

On $\sum (1/\sqrt{n})\cos (t_0 \log n)$ and $\sum (1/\sqrt{n})\sin (t_0 \log n)$, from a zero of $\zeta (s)$ of the form $s_0=(1/2)+it_0$

I assume (as hypothesis, for questions too) that $s_0$ a fixed (nontrivial) zero of the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty 1/n^s$ has the form $(1/2)+it_0$, Thus for a positive real ...
1
vote
0answers
45 views

How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All: I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com: ...
1
vote
0answers
37 views

Analytic continuation vs. series convergence near convergence boundary

Citing Wikipedia, the Riemann zeta function is the analytic continuation of $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ The series itself is only convergent in the right half complex plane ...
3
votes
1answer
51 views

Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same?

All: Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same ? They shall be all different, right ? Is there a proof of this statement ? Thank you.
2
votes
1answer
63 views

Dirichlet series expansion for Zeta(s)

Wikipedia lists a series expansion for $\zeta(s)$ here. How is the Dirichlet series below derived? I apologize in advance if this is a very simple question, I don't know much about Dirichlet series. ...
2
votes
0answers
47 views

Where to find Brun's original combinatoric treatment of Brun Sieve?

I tried to understand Brun's original combinatoric treatment of Brun Sieve. (Unfortunately, I do not understand German), so I could not read Brun's original paper as in following: Viggo Brun (1915). ...
2
votes
1answer
73 views

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis?

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis ? (Beside the obvious symmetry of Riemann Xi function, s <--> 1-s reflection) IMHO, at the ...
0
votes
0answers
32 views

What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
0
votes
0answers
38 views

Two-point correlations of the Riemann zeta function

In http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium/5, the author mentions that the function $$P(x) = 1-\left(\frac{\sin(\pi x)}{(πx)}\right)^2$$ seems to be, assuming the ...
2
votes
0answers
93 views

Irrationality of $\zeta(\frac{3}{2})$

Is $$ \zeta\left(\frac{3}{2}\right) = \sum_{n=1}^\infty \frac{1}{n^{3/2}} $$ an irrational number?
1
vote
2answers
63 views

What's the value of $\sum_{k=1}^{\infty}(\zeta(2k+1)-\zeta(2k+2))$?

I'm confused about this. I have this expression $$ \frac{1}{2}=\sum_{k=1}^{\infty}(\zeta(2k)-\zeta(2k+1)) $$ Now if I want claculate $\zeta(2)$ I'll do the apropriate manipulations to get $$ ...
2
votes
1answer
65 views

Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that ...
1
vote
0answers
45 views

Integral representations of $\zeta(s)$ using the floor/frac functions. How could this one be derived?

Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s ...
2
votes
3answers
69 views

Computing the tail of the zeta function $\sum_{n>x}n^{-s}$

I want to compute $$ f_s(x)=\sum_{n>x}n^{-s} $$ for some $s>1$ (in my case, $s=3$). Of course $$ f_s(x)=\zeta(s)-\sum_{n\le x}n^{-s} $$ but for $x$ large this is hard to compute. Are there good ...
8
votes
2answers
144 views

Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

Show that $$ L=\lim_{s\rightarrow\infty}\left(\int_0^s\frac{ds'}{\sqrt{s'}}-\sum_{s'=1}^s\frac{1}{\sqrt{s'}}\right) = 1.460\ldots $$ My attempts: To begin, rewriting the limit of the ...
0
votes
1answer
47 views

Is this infinite product for zeta(2) trivial?

I have crafted an infinite product for zeta(2) shown here. Euler's prime product is the only one I'm aware of. In checking Math World, I don't see any products. Is that because they are trivial?
4
votes
0answers
207 views

Approximate zeros of a (hypothetical) analog of $\zeta(s)$

[Added numbers 11/13.] Motivation (can skip). When prime powers $p_n$ are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say) $N= 30,$ $x>5$, at ...
16
votes
1answer
243 views

Is there any proof that the Riemann Zeta function is not elementary?

I'm just curious, has anyone ever proved that the Riemann Zeta function is not an elementary function? Here I am using the term "elementary" in the sense of Liouville or as defined in this paper. ...
5
votes
0answers
145 views

Connection between Riemann hypothesis and distribution of primes. [closed]

Honestly, I have to say that I have hardly any experience in number theory. That's maybe one additional reason why the Riemann hypothesis has such a "mystic" appearance for me. You always hear or read ...
2
votes
0answers
206 views

Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
12
votes
0answers
124 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
8
votes
3answers
198 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
9
votes
0answers
85 views

Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as ...
4
votes
2answers
114 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
3
votes
0answers
54 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
2
votes
1answer
64 views

Trivial zeroes of Zeta are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
5
votes
1answer
64 views

How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
1
vote
0answers
101 views

Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma ...
11
votes
1answer
237 views

Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th ...
3
votes
1answer
79 views

Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 ...
1
vote
0answers
44 views

Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with ...
2
votes
0answers
69 views

Please help me understand Analytic Density $\lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in A} \frac{1}{n^{\sigma}}$

$d (A) = \lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in B} \frac{1}{n^{\sigma}}$ for $B \subset \Bbb{N}$. So clearly this limit is $0$ for reciprocally summable (convergent) $B$. My goal ...
3
votes
4answers
249 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
8
votes
2answers
147 views

Series with $\zeta$

How do I calculate the following series: $$ \zeta(2)+\zeta(3)+\zeta(4)+ \dots + \zeta(2013) + \zeta(2014) $$ All I know is that $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. But this is not enough to ...
4
votes
0answers
69 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
2
votes
2answers
76 views

Abel's Summation formula help.

I want to be able to show that, \begin{equation} \sum_{p} \log(p) p^{-s} = s \int_{1}^{\infty} \frac{\theta(t)}{t^{s+1}} dt \end{equation} where $\theta(x) =\sum_{p \le x} \log p$. and $\theta(x) = ...
2
votes
0answers
40 views

Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...