Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [riemann-zeta]

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

0
votes
0answers
14 views

Using Euler's equation to calculate integral of $\zeta$ over the curve $Re(\zeta(t)=0)$ between first two zeros?

Is Euler's equation the right equation to use if I want to calculate a numerical value of $\int_{\{ {Re} (\zeta (t)) = 0 : 14.134 \ldots < {Im} (t) < 21.022 \ldots . \}} \zeta (t) d t$ It ...
5
votes
0answers
116 views

Evaluate $\int_0^1 \left[\log(x)\log(1-x)+\operatorname{Li}_2(x)\right]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]dx$

$$\mathfrak{I}=\int_0^1 \left[\log(x)\log(1-x)+\operatorname{Li}_2(x)\right]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]dx=4\zeta(2)\zeta(3)-9\zeta(5)\tag1$$ This integral ...
-3
votes
1answer
43 views

Proof that this theory is not true [on hold]

i^(2^s×3^s×5^s×7^s×11^s...) = 1 Let's assume that 2^s×3^s×5^s×7^s×11^s... = n, i^n = 1, On applying log to both sides, n x log(i) = 0, thus n = 0 or log i = 0, But log(i) = iπ/2 thus n = 0.
0
votes
0answers
52 views

What is the Explicit Formula for $\log(\zeta(s))$?

The explicit formula for the logarithmic derivative $\frac{\zeta'(s)}{\zeta(s)}$ is illustrated in (1) below. (1) $\quad\frac{\zeta'(s)}{\zeta(s)}=\frac{s}{1-s}+\log(2\,\pi)-\frac{1}{2}H_{\frac{s}{2}}...
0
votes
1answer
48 views

Approximate functional equation for the Riemann zeta function

The Riemann zeta function admits the approximation $$\zeta(s)\sim\sum_{n=1}^N\frac{1}{n^s}+\gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}},$$ in the critical strip, which is known as the approximate ...
3
votes
1answer
57 views

recurrence relation for $\zeta(2n)$

I found this formula. Is it correct? For $n\in\Bbb N,\ n\geq2$, $$\zeta(2n)=\frac{2n\pi^{2n}}{\Gamma(2n+2)}+\sum_{k=0}^{n-2}(-1)^{k-n}\frac{\pi^{2n-2k-2}}{\Gamma(2n-2k)}\zeta(2k+2)$$ Here's my proof....
5
votes
1answer
819 views

A line integral involving $\log \zeta(s)$

Let $\zeta$ denote the Riemann zeta function. Using the Cauchy integral theorem, can you evaluate $$I=\int_{\Re(s)=\frac{1}{2}} \frac{(2s-1)}{s^{2}(1-s)^2}\Bigg[\int \log((s-1) \zeta(s)) \mathrm{d}s\...
-1
votes
1answer
101 views

Questions on two Formulas for $\zeta(s)$

This question is related to the following two formulas for $\zeta(s)$. (1) $\quad\zeta(s)=\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^\infty\frac{1}{2^{n+1}}\sum\limits_{k=0}^n\frac{(-1)^k\binom{n}{k}}{(k+1)...
1
vote
2answers
58 views

Find the residue of $\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$ at $s=0$

Let, $\displaystyle f(s)=\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$. Prove that $Res(f,s=0)=A-\log x$ , for some constant $A$. At $s=0$ , $\zeta(s)$ has a pole of order $1$ and $\zeta'(s)$ has a ...
0
votes
0answers
53 views

An equation involving Non-Trivial Zeros of the Riemann Zeta function

$\rho$ is a Non-Trivial Zero of the Riemann Zeta function if and only if $$\displaystyle\int_1^{+\infty} \lfloor x\rfloor x^{-2-\rho} dx =\int_1^{+\infty} \lfloor x\rfloor \{ x \}x^{-2-\rho} dx $$ ...
0
votes
0answers
37 views

Irrationality of :$ \zeta \left(\frac{1}{\phi}\right)\Gamma{\left(\frac{1}{\phi}\right)}$

This number : $$ \zeta \left(\frac{1}{\phi}\right)\Gamma{\left(\frac{1}{\phi}\right)}$$ almost integer and it's close to $-3$ with $\phi$ is the Golden ratio and Zeta is the Riemann zeta function , $\...
0
votes
0answers
13 views

Means of powers of the zeta function

It is well known that the Lindel\"of Hypothesis is equivalent to the statement that $$\frac 1T\int_0^T|\zeta(1/2=it)|^{2k} =O(T^\epsilon)$$ for all positive integers $k$ and all positive real $\...
3
votes
1answer
445 views

Why couldn't Baez-Duarte prove the Riemann Hypothesis?

Define \begin{equation} I_n=\int_{0}^{1/n} |U s_{n}(x)|^2 \mathrm{d}x \end{equation} where $Us_{n}(x)=\frac{1}{x}\sum_{j=1}^{n} \frac{\mu(j)}{j}\rho(jx), \mu$ denotes the Mobius function and $\rho(y)...
1
vote
0answers
58 views

On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
1
vote
1answer
82 views

A line integral involving $\zeta(s)$

Consider the line integral $$I=\int_{1/2 -i\infty}^{1/2 + i\infty} \frac{\log((s-1)\zeta(s))}{s} \mathrm{d}s-\int_{1/2 -i \infty}^{1/2 + i\infty} \frac{i\arg \zeta (s)}{s}\mathrm{d}s$$ where $\zeta$ ...
5
votes
4answers
102 views

A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$
0
votes
0answers
32 views

Question on Probability Distributions Related to the Riemann Xi Function $\xi(s)$

This question is related to the $g_i(x)$ functions which are defined below both in expanded form and in terms of the $f_i(x)$ functions defined in my previous question at ref(1) where all $f_i(x)$ ...
4
votes
1answer
79 views

Simple Analogy to explain $\sum_{n=1}^\infty n = -1/12$

I'm looking for a simplified analogy to explain why the following formula does not actually mean what it seems to mean: $\sum_{n=1}^\infty n = -\frac{1}{12}$ I get this question all the time from ...
1
vote
1answer
45 views

On the behavior of $|\zeta(1/2 + it)|$ for $t\in(-14,14)$

Denote by $\zeta$ the Riemann zeta function. Since $\zeta(1/2 + it)\neq 0$ for $|t|\leq 14$, it seems to follow that $|\zeta(1/2 + it)|$ is either strictly decreasing or strictly increasing on $(-14,...
-1
votes
0answers
16 views

Analogue of the Mertens function for the extended Riemann conjecture

It is known that the Riemann conjecture is equivalent to $$M(x) = O(x^{\frac12+\epsilon}),$$ where M(x) is the Mertens function. Does there exist an analogue to this equivalence for the extended ...
1
vote
0answers
34 views

Need someone to show me how the Zeta function is equal to Euler's product formula?

$\zeta(x)$=$1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+\frac{1}{6^x}+\cdots$Euler's product formula states, for all $x$ greater than $1$ we have:$\zeta(x)$=$1\over 1-\frac{1}{2^x}$$\...
3
votes
0answers
73 views

Evaluation of :$\sum_{n\geq 2}\frac1n\Gamma(\frac1n)^{\zeta{(\frac1n)}}$

I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form $$\sum_{n\geq 2}\dfrac1n\left(\Gamma\left(\...
1
vote
0answers
63 views

A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
-4
votes
1answer
60 views

What is the value of $\sum\limits_{k=1}^{\infty}\frac{1}{k^3}$? [closed]

What is the value of $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^3}$ or namely the Riemann Zeta function for $s=3$, $\zeta(3)$
0
votes
0answers
9 views

Arbitrarily Close Facsimile of a Patch of an Arbitrary Function

In my opinion, the most amazing theorem in the whole of mathematics is that if you have an arbitrary function over the complex plane, and you take a patch of this function that is free of zeros and ...
0
votes
0answers
85 views

Questions related to the Riemann zeta function where $|\zeta(s)|=|\zeta(1-s)|$

The Riemann Zeta functional equation is defined as follows. (1) $\quad\zeta (s)=f(s)\,\zeta(1-s)\,,\quad f(s)=2^s\pi^{s-1}\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma (1-s)$ Note that $|f(s)|=1$ along ...
0
votes
1answer
140 views

Is Proposition 5.1 in Choie et al 2007 Correct?

Preliminaries The focus of Choie et al 2007 (references below) is Theorem 1 in Robin 1984 which states that the Riemann Hypothesis is true if and only if there are no numbers $\gt$ 5040 that violate ...
1
vote
0answers
45 views

Riemann zeta zeros Fourier like divergent square wave. Can you complete this analogy?

The question is to complete this analogy: $$\left|Z(t)\right|=\left|\zeta \left(\frac{1}{2}+i t\right)\right| \tag{1}$$ is to: $$Z(t)=e^{i \vartheta (t)} \zeta \left(\frac{1}{2}+i t\right)...
0
votes
0answers
56 views

Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$ (2) $\quad f(x)=\sum\...
0
votes
0answers
37 views

What are the Fourier Transform of Dirichlet L-function?

We know that the Fourier transform for Riemann $\Xi$ ${\displaystyle \Xi (t)=\xi ({\frac 12}+it) }$ where: $\xi (s)={\tfrac {1}{2}}s(s-1)\pi ^{{-s/2}}\Gamma \left({\tfrac {1}{2}}s\right)\zeta (s)...
0
votes
0answers
51 views

Interesting zeta-like series

So this idea was inspired by a post I saw quite a while back asking about the convergence of the series $\sum_{n=1}^\infty \frac{1}{n^{1+|\sin n|}}$ (to which I actually still don’t know the answer). ...
0
votes
0answers
47 views

Anomaly in probability of 3 randomly-selected integers being coprime

The probability that two randomly-selected integers will be coprime is given by $[\zeta(2)]^{-1}$, where $\zeta$ is the Riemann zeta function. Similarly, for three such integers the probability is $[\...
5
votes
2answers
154 views

Growth of Digamma function

For $1\le \sigma \le 2$ and $t\ge 2$, $s=\sigma+it$ prove that $\displaystyle \frac{\Gamma'(s)}{\Gamma(s)}=O(\log t)$. From Stirling's formula we have, $\displaystyle \Gamma(s)\approx \sqrt{2\pi}\exp\...
4
votes
2answers
533 views

What is this sum equal to? $\sigma(n)=\sum_{i\neq j} \frac{1}{i^n j^n}$

I have recently come across the following sum, taken over all positive integers $i$ and $j$ such that $i \neq j$: $$ \sigma(n)=\sum_{i\neq j} \frac{1}{i^n j^n}, $$ where $n$ is a positive integer ...
3
votes
0answers
83 views

Questions related to $f(x)$ where the Riemann Xi function $\xi(s)=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx$

I realize this question is a bit long and contains quite a few formulas, but I believe a considerable amount of background and context are needed to fully understand my questions below. Also, I ...
4
votes
1answer
107 views

Proving an identity for $\sum_{m,n\in\mathbb{Z}_{>0}}\frac{\gcd(m,n)^r}{m^sn^t}$.

My task is to prove the well-known identity Here all variables are positive integers I only know I should use mobius inversion formula, but how to proceed I am getting confusion, please any one ...
1
vote
2answers
73 views

What does $1+1/8+1/27+1/64+1/125+(1/n)^3$ equal to?

I'm curious of what does this sum: $1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}+\frac{1}{216}+...+(\frac{1}{n})^3$ or the Riemann zeta function: $\zeta({3})$ approach. I watched a few ...
6
votes
0answers
99 views

For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?

Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason? Claim. ...
1
vote
0answers
86 views

Abscissa of convergence $\sum_{n=0}^{\infty}\frac{\mu(n)}{n^s}$

I have seen statements like $\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$ is convergent for $\Re(s)>1$, and I have seen proof of it being zero (and therefore convergent) when $s = 1$ but haven’t seen ...
1
vote
1answer
53 views

Rational functions over $\mathbb{Z}$ yielding series that converge to a given real number

I wonder what can be said about the set of rational functions over $\mathbb{Z}$ for which summation of the values at positive integers converges to a given real number. More precisely, this is about ...
19
votes
1answer
429 views

Is there an integral for $\frac{1}{\zeta(3)} $?

There are many integral representations for $\zeta(3)$ Some lesser known are for instance : $$\int_0^1\frac{x(1-x)}{\sin\pi x}\text{d}x= 7\frac{\zeta(3)}{\pi^3} $$ $$\int_0^1 \frac{\operatorname{...
0
votes
0answers
23 views

Riemann hypothesis for $L(s,\chi)$ and $L(s,\chi^\sigma)$

If $\sigma \in \text{Gal}(\mathbb{Q}(\zeta_{\infty})/\mathbb{Q})$ do we know or expect that two Dirichlet L-functions $L(s,\chi)$ and $L(s,\chi^\sigma)$ have more in common, especially in term of ...
2
votes
1answer
56 views

Is Riemann's R-function $R(x)$ differentiable?

Edit note: I mis-wrote the expression originally, resulting in helpful-but-incorrect feedback. The function is now correctly expressed. I suspect that the answer to this is a resounding 'no', but I'm ...
0
votes
1answer
33 views

Order of $\Gamma(n/2)$

I want to estimate the order of $\Gamma(n/2)$. We have from Stirling interpolation , for sufficiently large value of $n$, \begin{align} \Gamma(n/2)&\approx \sqrt{\frac{4\pi}{n}}\left(\frac{n}{2e}...
6
votes
4answers
109 views

If $ \int_{0}^{1} \frac{\ln x}{1-x^2} dx = -\frac{π^2}{\lambda} $ find $\lambda$ given that $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{π^2}{6} $

If $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{π^2}{6} $$ then $$ \int_{0}^{1} \frac{\ln x}{1-x^2} dx = -\frac{π^2}{\lambda} $$ then the value of $\lambda$ equals? My attempt- I tried using ...
0
votes
0answers
27 views

A function that satisfies $n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$

I have been trying to find a function $f$ that yields the following: $$n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$$ where $f(n)\to1$ and $f(n)\gt1$ for all sufficient $n$. I suspect that Mertens ...
3
votes
1answer
42 views

Proof of $ \zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$

Prove that $\displaystyle \zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$,near $s=1$, where $\gamma$ is Euler's Constant. I've proved $\displaystyle \zeta(s)=s\int_1^{\infty}\frac{[x]-x+1/2}{x^{s+1}}\,dx+\frac{...
0
votes
0answers
23 views

Superscript numbers on a Hurwitz Zeta function (Wolfram Alpha)

http://www.wolframalpha.com/input/?i=x(n%2B1)%3Dsqrt(n%2F(n%2B2))*x(n) See the x(n) solution given - there is a Zeta function with a (0,1) superscript, which I'm unfamiliar with. Additionally, the ...
0
votes
0answers
36 views

What is asymptotic and error bound for $\sum\limits_{k=1}^K\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)$ as a function of $K$?

This is a follow-on of my previous question What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$? My previous question was related to the following two formulas. (1) $\...
0
votes
1answer
38 views

How can I find the number equaling 2 in Riemann's zeta function?

I used some graphs but did not find the correct value, $$\sum_{n=1}^ \infty \frac{1}{n^x} = 2 $$ what is the x value?