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

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60 views

Why do you need to prove the error term goes to zero for the complete derivation of the Euler Product Formula?

I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I ...
0
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0answers
27 views

What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le c_k Y^k(t)$

From the Vinogradov-Korobov estimate, we have $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \ll Y^k(t)$ What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le ...
4
votes
3answers
229 views

How to solve this integral: $\int_{-\infty}^\infty\frac{x^2 e^x}{(e^x+1)^2}\:dx$

I am trying to solve an integral like this: $$ I=\int \frac{x^2 e^x}{(e^x+1)^2} dx $$ And I get this answer: $$ \int \frac{x^2 e^x}{(e^x+1)^2}dx=x^2-\frac{x^2}{e^x+1}-2x\text{ln}(e^x+1)+2\int\text{ln}(...
1
vote
1answer
86 views

Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
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1answer
57 views

How to calculate non-trivial zeros of the Riemann zeta function

I wanted to know how riemann calculated some non-trivial zero of the zeta function. Would I like a manual calculation.
3
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2answers
113 views

Riemann Zeta Function integral

I was reading about the Riemann Zeta Function when they mentioned the contour integral $$\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x - 1} dx$$ where the path of integration "begins at $+\infty$, ...
2
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1answer
158 views

Proof of Riemann Hypothesis

This proof was released this year: http://arxiv.org/abs/1508.00533 Where is the mistake? I just found it and was wondering how obviously wrong it is.
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1answer
140 views

Average number of square free divisors for $n\leq x$

Let $d_{sf}(n)$ be the number of square-free divisors of $n$, and let $D_{sf}(k)=\sum_{n=1}^{k} d_{sf}(n)$ denote the corresponding summation function. Mertens showed that the asymptotic expansion of ...
6
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0answers
99 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
2
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1answer
64 views

What is the effective lower bound on gaps between zeta zeros?

In this question here: Upper bound on differences of consecutive zeta zeros by Charles it is said that: "There are many papers giving lower bounds to: $$\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}$$ ...
2
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0answers
48 views

Easy computations from a representation for Riemann zeta function, and Riemann zeta zeroes on critical line

The following series for Riemann Zeta function converges for $\Re s>0$ $$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}).$$ See for example this site or Wikipedia [1]. ...
1
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1answer
33 views

Breaking up integral representations by convergence

A known integral takes the form of $$\zeta(3)=\frac{1}{2}\int_{0}^{\infty} \frac{t^2}{e^t-1}dt$$ Through Wolfram part of the integral converges to $$\int_{0}^{\infty} \frac{t}{e^t-1}dt = \frac{\pi^...
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0answers
34 views

Can you rewrite this function to be independent of the variable $p$?

A while ago Roger L. Bagula came up with the idea to study this function: ...
2
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3answers
127 views

Evaluating an infinite sum involving possibly hypergeometric terms

I was considering the following infinite sum $$ A(n) = \sum_{k=2}^{\infty}\left[\frac{(-1)^{k+n-1}}{k^n}(0k -1)(k-1)(2k-1)...((n-1)k-1) \right] $$ Some cases: $$ A(1) = \sum_{k=2}^{\infty}\left[\...
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0answers
127 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
0
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1answer
62 views

Is this plot of the argument of the Riemann zeta function around ZetaZero(127) correct?

(*Mathematica 8 start*) Clear[n, k, t, z, FL, NZ] N[ZetaZero[127]] NZ[t_] = Arg[Zeta[1/2 + I*t]]/Pi; Plot[NZ[t], {t, 280, 284}] Plot[NZ[t], {t, 282.3, 282.6}] ...
2
votes
1answer
60 views

Generating function of Riemann zeta function

I want to know about the generating function of the Riemann zeta function which is related with the Laurent expansion at $z=0$. $f(z) := \dfrac{d}{dz} \log(\sin\pi z)$ $f(z) = \dfrac{1}{z} -2\sum \...
0
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0answers
47 views

$\sum \zeta (s)$ converging proof?

I was pondering about the fact that the Zeta function can be represented as an infinite sum, $\zeta (s) = 1/1^s + 1/2^s + 1/3^s +\ ...$, and I thought about the infinite sum of the zeta function ...
2
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1answer
57 views

An algebraic manipulation of the Zeta function

Consider the following form of the Riemann Zeta function: $s\in\mathbb{C}$, such that: $\left |s \right | > 1$ $\zeta \left ( s \right )= 1+2^{-s}+3^{-s}+4^{-s}+5^{-s}...$ Now, due to the ...
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1answer
60 views

Could someone explain how the Gram series relates to Riemann's function?

I was reading an article on the distribution primes which mentions the following equation for Riemann's function $R(x)$: $$R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n}\text{li}(x^{1/n}) = 1 + \sum_{k=1}...
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1answer
57 views

How to calculate $\zeta(s)$ using residues?

We have the Riemann zeta function $$ \zeta(s) = \frac{\pi}{2} \int_{\gamma} \frac{dz}{2\pi i} \, \frac{1}{z^s} \cot(\pi z) $$ and I want to use the Residue theorem in order to understand why it is ...
3
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1answer
38 views

asymptotic expansion in powers of $ 1/s $

How could I get the coefficients in the Dirichlet series expansion $$ \frac{\zeta ' (s)}{\zeta(s)}= \sum_{n=1}^{\infty} \frac{a(n)}{n^s} $$ for the logarithmic derivative of the Riemann zeta ...
0
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0answers
40 views

Problem inside the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$

I am having trouble with one step for the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$ found here. In the below steps, how do we get from $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
11
votes
2answers
352 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
9
votes
1answer
191 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
0
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0answers
19 views

Importance of the real-rooted asympototics of $f_n(z)$ that uniformly converges to Riemann $\Xi(z)$ function

We are learning Riemann $\Xi(z)$ and Riemann $\zeta(s)$ functions. This question is related to an earlier one. (1) Suppose that a family of functions, $f_n(z)$, uniformly converges to Riemann $\Xi(z)$...
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3answers
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Are the nontrivial zeroes of the Riemann zeta function countable?

It is known that the set of non trivial zeros is an infinite set. But is it known if it is a countable, or uncountable infinite set?
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0answers
140 views

Assuming the Riemann hypothesis, does this integral give the Riemann zeta zeros when increasing Working Precision in Mathematica?

It is probably well known that the Riemann zeta zeros satisfy the following equation: $$\frac{\arg \left(\zeta \left(\rho _n+\frac{1}{1000000000000000}\right)\right)}{\pi }+\frac{\vartheta \left(\Im\...
2
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0answers
70 views

Is there a contiguous locus of the equality $a = b$ in $\zeta(\rho + \varepsilon)=a + î b$ in the near of a root?

This is just by an accidental couriosity: In the near of a root $\rho$ of the Riemann's zeta - can there be a continuous line starting from $\zeta(\rho)=0$ to $\zeta(\rho + \varepsilon_j)=a_j + î b_j ...
2
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1answer
219 views

Argument of the Riemann zeta function on Re(s)=1

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some ...
3
votes
2answers
65 views

Clausen zeta function

For $0 < \theta < 2\pi$, define $$\kappa(x,\theta) = \frac{1}{\zeta(x)}\sum_{n=1}^\infty \frac{e^{ in\theta}}{n^x}$$ for $\Re(x) > 1$. It is easy to see that $$\kappa(x,\theta) = \frac{1}{\...
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0answers
29 views

Values of Riemann zeta function at small height along Re(s)=1

I am looking for sources which give numerical and theoretical computations of $\zeta(1+it)$ and the completed zeta function (without the $s(s-1)$) $$\xi(1+it)=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$$ ...
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0answers
95 views

When does linear combination of real-rooted entire functions of genus 0 or 1 remain real-rooted?

In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions: $$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$ where $b&...
4
votes
1answer
80 views

On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt \left(\sum_{n=1}^{\infty}\frac{1}{n^{...
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1answer
41 views

what is the value of Chebyshev function at non-integer value?

What is the value of Chebyshev $\psi(x)$ function at non-integer values ? For example, what is the value of $\psi(3.56)$? I have seen, in same place, it seems that $$\psi(3.56)=\psi(3)$$ And in ...
0
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0answers
48 views

Summation continuation, over no common divisors

Recently i've thought of a new step in my summations. I've been mentioning them in my previous questions. And i know there are certain point were it doesn't work. But most of the time it does. $$\...
0
votes
1answer
144 views

Relationship between perfect squares and infinite series (zeta function)

I noticed something when scribbling the zeta function - $$\zeta(3) = \sum_{n=1}^{\infty}\frac{1}{n^3} = 1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \frac{1}{125} + \frac{1}{216} + \frac{1}{343} + ...
2
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2answers
338 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
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1answer
114 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 ...
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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
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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
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1answer
100 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 \...
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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 ...
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0answers
97 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
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0answers
67 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 ...
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0answers
32 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
114 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
121 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
59 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 \...