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

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1
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1answer
99 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
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0answers
38 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, ...
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0answers
31 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 ...
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0answers
74 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
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1answer
104 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
61 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
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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
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1answer
126 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
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0answers
63 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
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1answer
89 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
159 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
65 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
125 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
248 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
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1answer
61 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
157 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
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0answers
58 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 ...
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0answers
75 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
37 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
99 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
746 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
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0answers
59 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
190 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
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0answers
114 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
94 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
284 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
141 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
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1answer
204 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
269 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: ...
38
votes
2answers
2k 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
119 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
63 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
145 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
211 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 ...
19
votes
2answers
508 views

Formula for $\zeta(3)$ -verification

By simple manipulating with some series I have found the following formula for $\zeta(3)$: $$\zeta(3)=\frac27\sum_{k=0}^{\infty}(-1)^kB_{2k}\frac{\pi^{2k+2}}{(2k+2)!},$$ where $b_k$ are Bernoulli ...
3
votes
1answer
133 views

Closed form for $\sum\limits_{k=1}^{\infty}\zeta(4k-2)-\zeta(4k)$

I am looking for a closed form of the expression $$ \sum_{k=1}^{\infty}\zeta(4k-2)-\zeta(4k) $$ Closed form would be something in terms of constants such as $\pi$, $\gamma$, $e$, etc.
5
votes
1answer
157 views

Is this closed form expression of $\zeta(3)$ correct?

In this paper on pages 6 & 7. Page 6 lists the variables used in the equation on page 7. The author claims a closed form expression of $\zeta(3)$ (he also goes on to claim a closed form ...
5
votes
0answers
87 views

Is there a name for this constant?

$$\prod_{n=2}^\infty \zeta(n)=2.294856591673313794183$$ Is there a name for this constant and what are some if its properties?
12
votes
4answers
465 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
15
votes
2answers
312 views

Proving that $\left(\frac{\pi}{2}\right)^{2}=1+\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k-1}}$.

Wolfram$\alpha$ says that we have the following identity $$ \left(\frac{\pi}{2}\right)^{2}=1+2\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k}} $$ but, how does one prove such identity?
2
votes
2answers
122 views

Unconventional way, how to expand to Maclaurin series

Let's have function $f$ defined by: $$f(x)=2\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}-x\sum_{k=1}^{\infty}\frac{e^{kx}}{k^2},\quad x\in(-2\pi,0\,\rangle$$ My question: Can somebody expand it into a ...
11
votes
2answers
222 views

Evaluating $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$

I want to find $$\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$$ I let $f(z)=\sum_{n=1}^\infty\frac{1}{2^{3n}}z^{3n}$ and now $$\sum_{n=1}^\infty ...
3
votes
3answers
156 views

$(1-2^{1-s})\zeta(s)$ is an entire function

Show that $(1-2^{1-s})\zeta(s)$ is an entire function, which is represented by the series $$(1-2^{1-s})\zeta(s)=1-\dfrac{1}{2^s}+\dfrac{1}{3^s}-\dfrac{1}{4^s}+\cdots$$ for $\Re{s}>1$. From the ...
13
votes
1answer
243 views

Prove $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure. Mathematica can provide a closed form for ...
14
votes
2answers
301 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
1
vote
1answer
57 views

Series divergence - how precise should the answer be

Morning. I've written down some of my reasoning and arguments as to why the series diverges, however I am not certain I can safely conclude it diverges to $\infty$. Would you give it a look, please? ...
3
votes
0answers
83 views

How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
3
votes
2answers
41 views

Limit $(t-1)\zeta(t)$ as $t\rightarrow 1^+$

Show that $\lim_{t\rightarrow 1^+}(t-1)\zeta(t)=1$. For $t>1$, we can use the definition $\zeta(t)=\sum_{n=1}^\infty \dfrac{1}{n^t}$, so it is approximately $\int_1^\infty \dfrac{1}{x^t}dx$. ...
1
vote
1answer
65 views

Reflections of zeros of zeta function in the critical strip

Show that if $a$ is a zero of the zeta function in the critical strip, then so are $\bar{a}$, $1-a$, and $1-\bar{a}$. The definition of $\zeta$ is ...