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

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Equating two definitions of Zeta function

The Zeta function $\zeta(s)$ is defined as following $$ \zeta(s)=1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\frac1{5^s}+\cdots=\sum_{n=1}\frac1{n^s} $$ Now it has been shown that $$ \tag1 ...
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Eulers proof sum of natural numbers

I've to recheck Eulers proof of the sum of the natural numbers, but I dont now exactly what it is? It has something to do with the $\zeta(s)$? Thanks in advance
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A question from Titchmarsh's zeta function book.

On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that: $\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get ...
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Bernoulli Numbers generating function and Riemann Zeta function

I've been studying Bernoulli numbers and I came across this summation: $$ \sum_{n=1}^{\infty}\frac{B_n x^n}{n!} = \sum_{n=1}^{\infty}\frac{-n \zeta(1-n) x^n}{n!} = -\sum_{n=1}^{\infty}\frac{\zeta(1-n) ...
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Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...
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How shall I get an estimate of $\int_{1-\frac{c}{log t}-iT}^{1-\frac{c}{log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$?

Please help me on the following. I need to estimate $$\int_{1-\frac{c}{\log t}-iT}^{1-\frac{c}{\log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$$ where $c$ is a constant, $T>0$. What i ...
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When is $\zeta(s)=0$?

At what real constant does $\zeta(s)=0$? Does that constant have any significance? Thank you very much for any help provided.
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Is this Riemann Zeta function integral formula known about?

I discovered that $$\zeta(s)=\int_0^1\frac{(-\log(1-x))^{s-1}}{x(s-1)!}dx.$$ Is this an obvious result that is not worth much interest or is this new and unique?
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Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$\ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
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Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
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Prove that these Dirichlet L function are equal to these zeta function products.

Prove or disprove that: $$2 L_{2,1}(s)-2 \zeta (s)+\zeta (s)=\left(1-\frac{1}{2^{s-1}}\right) \zeta (s)$$ $$3 L_{3,1}(s)-3 \zeta (s)+\zeta (s)=\left(1-\frac{1}{3^{s-1}}\right) \zeta (s)$$ Where ...
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66 views

How $\zeta(-1)$ is defined? [duplicate]

I know some proof that $\zeta(-1)$ equals to $-\frac{1}{12}$. here it is: Let $S_1 = (1)+(-1)+(1)+(-1)+...$ Then we have that $2S_1=1$ because if we shift the second $S_1$ for 1 to right we have ...
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Identity for $L(s,\chi)L(s,\bar\chi)$

I was told recently that there is an identity roughly of the form $$L(s,\chi)L(s,\bar\chi)=\zeta(s)^2$$ If true, it seems like there should be a short proof of this. Could someone supply a ...
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Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
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$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
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Applying the inversion formula to Hardy's theorem

So, in the proof of Hardy's theorem, who says that $\zeta$ has infinite zeros on the critical line, we have eventually that ...
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What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
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Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
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How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?

In wolfram alpha the values of $$\zeta(0.5+ie^x)$$ closed to zero then How do I know the real values of $\zeta(0.5+ie^x)$ for large real number $x$ ? Thank you for any help
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Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
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How to get sine term in Analytical continuation of $\zeta(s)$

I am able to prove the symmetric functional equation that Riemann gives in his paper, using Poisson Summation and properties of $\theta(x)$. The functional equation is given like so, ...
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One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
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Does this product over the primes converge, and if so, to what?

I've been trying to play around with the product: $$\prod_{p \text{ prime}}\frac{1}{1-(-p)^{-1.5}}$$ Where the product runs over all the prime numbers. The product is similar in appearance to the ...
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Growth of the zeta function on the line $Re(s)=\frac{1}{2}$

I've seen that on the line $Re(s)=\frac{1}{2}$, $\zeta(s)=O(t^{\frac{1}{4}})$ where, as usual, $s=\sigma+it$. My teacher has told me that this can be derived directly from the functional equation of ...
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Estimating $\log|\zeta(\sigma+it)|$ for $\sigma$ sufficiently large

In a paper I am reading, I've come across the estimate $$ 2\pi\sum_{\substack{\sigma<\beta<\sigma_0\\ T<\gamma\leq ...
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Limit involving Zeta and Gamma function

Can someone help me evaluate this limit? $$\lim_{x\to +\infty}\frac {\zeta(1+\frac 1x)}{\Gamma(x)}$$ I never came across this kind of limit so I don't even know where to start.
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Limit of zeta function in $x = 1$

How can I prove that $\lim_{x \rightarrow 1}{\sum_{n=1}^{\infty}{\frac{1}{n^x}}} = \infty$? My idea is to show that we can exchange the positions of limit and sum, obtaining the harmonic sum, that we ...
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Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula

We have the relationship $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ Now I know one uses the Poisson summation formula to prove this. The Poisson summation formula comes from Fourier Transform ...
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Solving summation; ( double sum).

I found the expression to the sum of powers long ago and ofcours I think it is true but i don't know for sure, the problem is, it's little though for me to test and try it out. Also i'd like to know ...
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Proving an inequality, involving the Riemann-$\zeta$-function

I'm studying a book, which wants to prove that for some constants $C,\kappa$, we have $$\left|\frac{\zeta ''(s)\zeta (s) -2\zeta '(s)}{\zeta (s)^2}\right|\le C*|t|^\kappa~~~~,\text{for ...
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Limit approach to finding $1+2+3+4+\ldots$

When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit: ...
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Zeta regulated product, solving without the zeta function.

Earlier i've asked about how to calculate divergent products, i got some directions which made me curious. Now i'm wondering is this correctly done. Divergent products. The most commen divergent ...
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What is the power series expansion for Riemann-Zeta at $0$?

What are the first few terms of the Laurent series expansion of $\zeta(0)$? It gets mentioned here but they only show the first term and I am kind of confused on how they got $-1/2$.
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Ways to prove Eulers formula for $\zeta(2n)$

I recently, out of interest, tried to prove Euler's formula $\zeta{(2n)}=(-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!}B_{2n}$ for all $n\in\mathbb{N}$. I adapted Euler's original proof for ...
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Euler product of $\sum (2^k n + 1)^{-s}$

do we know for a given $k > 2$ the Euler product of $\ \displaystyle\sum_{n=0}^\infty (2^k n + 1)^{\textstyle-s} \ $ ? I saw that every prime numbers will appear in it, as well as some non-prime ...
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Can $\zeta(s)$ be written in the form $\zeta(s)=\Re(\zeta(s))+i·\Im(\zeta(s)) $ for some subset of $\mathbb{C}$?

Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like... ...
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Other way show that $\zeta(-2)=(-\frac{1}{12})\mod 2$? [closed]

Lemma: We knew that for any integer $a$ : ${a}^{p}=a \mod p$. Then $1^p=1\mod p ,2^p=2\mod p ,3^p=3\mod p , \ \dots,\ n^p=n\mod p $. Just to sum each term by term RHS and LHS we will get the ...
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Why $\zeta(-2) $ is not $\sum_{n=1}^{\infty}\frac{1}{n^{-2}}$? [duplicate]

Let $\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}$ a standard formula. I'm confused if you tell me: does this series: $\sum_{n=1}^{\infty}\frac{1} {n^{s}}$ converge? I will answer you: this series ...
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Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?

From this question, it is easy to derive that a zero of $\xi(a+s)\pm \xi(a+1-s)$ should occur when: $$\displaystyle{\frac {\zeta \left( s+a \right) }{\zeta \left( s-a \right) }}=\pm{ \frac {{\pi ...
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Alternating series involving zeta function

Can anyone help me attain the result for the following series? $$\sum_{n=2}^{\infty} \frac{(-1)^n \zeta(n)}{n(n+1)}= \frac{1}{2} \left( \log 2 + \log \pi +\gamma -2 \right)$$ I don't know how to ...
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Riemann zeta function, representation as a limit

is it true that $$\displaystyle \zeta(s) = \ \lim_{\scriptstyle a \to 0^+}\ 1 + \sum_{m=1}^\infty e^{\textstyle -s m a } \left\lfloor e^{\textstyle(m+1)a} - e^{\textstyle m a} \right\rfloor$$ my proof ...
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can you integrate the dirac delta over any domain?

At eq 28 the following author takes it from 1 to infinity. I've never seen this before and i'm not sure his equation is correct. ...
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Would the Riemann Hypothesis being false affect how frequently primes occur in the number system?

I want to know that if Riemann hypothesis is false (big assumption) would that lead to any effect in how frequently primes occur . Well I got this half cooked information from here: ...
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Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
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Showing a series based on the Riemann zeta function converges (uniformly)

This question comes from Lars Ahlfors' complex analysis (page 178). We define $\zeta(z) = \sum_{n=1}^{\infty} n^{-z}$. This is just the Riemann zeta function. I am struggling however to prove that ...
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infinity series of Riemann zeta function at odd integers

Properties of Riemann zeta function at odd and even integers diverge dramatically, which can be proved by many evidences. I once found an infinity series in wikipedia, it reads $$ ...
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Sum of reciprocals of primes diverges

I can show that $$\log(\zeta (s)) = \sum _{p\in\Bbb P} \frac{1}{p} + R(s)$$where $$R(s) = \sum _{m\geq 2} \sum_{p\in\Bbb P} \frac{1}{m} \frac{1}{p^{ms}}$$ where $\Bbb P$ is the set of all primes, ...
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How the steps followed help me to understand

Few days before I posted one problem on MSE here in which it was asked to determine the residue. I understood almost entire but got stuck the following Note that after step 4, in step 5 it is ...
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What is the Möbius analoge for Ihara's $\zeta$ function?

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have $$ ...
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Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...