# Questions tagged [riemann-zeta]

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

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### How to evaluate $\sum_{n=2}^{\infty}\left(\frac{1}{\zeta (n)}-1\right)$?

It is well known and easy to evaluate, that $\sum_{n=2}^{\infty}\left(\zeta (n)-1\right)=1$. I am trying to evaluate $$\sum_{n=2}^{\infty}\left(\frac{1}{\zeta (n)}-1\right)$$ The classical way gives ...
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### Empirical study of zeroes of RIemann Zeta Function [closed]

Has there been any study done on how the imaginary part of the zeroes are distributed ? Is there a closed form for a first zero ? is there a generating function ?
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### Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$ [closed]

It seems that the above identity is true. Can this be proven? Or are there references treating sums like the right hand side? The above constants, $\gamma_{n}$, are the Stieltjes constants. Thanks.
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### If $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x)$ is true, what does that say about the Riemann zeta zeros?

Let $\pi(x)$ be the prime counting function. The statement $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x)$ is in "essence" weaker than what we can conclude from the RH if the RH is true. I assume it ...
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### Why do fractal-like patterns appear in this sequence?

I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself. It caught my ...
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### For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges

[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)] Let $f(n)$ be an arithmetical function which is periodic mod $k$. If $$\sum_{r=1}^k f(r)=0$$ then prove that the ...
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### "Mollifier" of the Dirichlet L-function

I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
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### Proofs involving manipulation of divergent series

Is this proof valid even though the harmonic series it is based on is divergent? Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$ Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
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### $0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?

Define $f(s)$ as $$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$ where we take the upper complex plane as everywhere analytic. Notice this is an antiderivative of the Riemann Zeta function, ...
1 vote
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### Question concerning an assertion regarding the modulus of the Riemann Zeta function (follow up)

Update Nov 25th-- I made a few small changes, especially in my assertion of "Lemma 1". I have yet to find the final series of telescoping and collapsing summands described at the end below. ...
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### Series involving derivative of Riemann Zeta function: $\displaystyle \sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$

1. Question Could anyone recommend a useful method for approaching the following series? $$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$ Where $\zeta(z)$ is the Riemann Zeta function. I've seen that there are ...
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### Convergence of $\sum_{\rho} |\Gamma(\rho)|$

In his proof of the prime number theorem Littlewood claims that the sum $\sum_{\rho} |\Gamma(\rho)|$ converges, where ${\rho}$ ranges over the non-trivial zeros of $\zeta$ counting multiplicity. I ...
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### What's so special about the Riemann Zeta zeros being on the $x=\frac{1}{2}$ line?

There are lots of explanations (videos, blogs) on the Riemann Zeta function online. They all describe analytic continuation beyond the $x=1$ line and then the fact that there are trivial zeros and ...
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### Question concerning an assertion regarding the modulus of the Riemann Zeta function

Posting this question here as the community has addressed the Riemann Zeta function with questions such as, i.e. with Simpler zeta zeros and Zeta function zeros and analytic continuation. Define the ...
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### Functional equation for $\theta$ function using functional equation of $\zeta(s)$

Let $$\theta(t) = \sum_{n \in \mathbb Z}e^{-n^2 \pi t}.$$ We can derive a functional equation for $\theta$ using Poisson summation formula: $$\theta(1/t) = \sqrt t \theta(t).$$ Riemann uses the above ...
1 vote
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### Asymptotic Behavior of Riemann Zeta Function

I am currently reading an article that claims that the value $$\inf \{ a \in \mathbb{R}, |\zeta(\sigma + it)| = O(|t|^{a}) \}$$ is zero if $\sigma > 1$ and is $\frac{1}{2}-\sigma$ if $\sigma < 0$...
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### Using Riemann's methods to derive the floor function of x from the zeta function

I thought it would be interesting to try and use Riemann's methods to derive (rather than the quantity of primes less than $x$) the quantity of natural numbers less than $x$. This function is, of ...