Questions tagged [riemann-zeta]

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

If $s$ is a complex number for which $\Re s > 1$, the infinite series

$$\sum\limits_{n = 1}^{\infty} \frac{1}{n^s}$$

defines an analytic function in the domain $\{s : \Re s > 1\}$, and can in fact be extended to $\mathbb{C} \setminus \{1\}$; this extension is called the Riemann zeta function:

$$\zeta(s)=\frac{\Gamma(-s-1)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{s-1}}{e^x-1}dx$$

where the contour travels from $+\infty$ on the $x$-axis to a counter-clockwise circle around the origin, and back to $+\infty$ on the $x$-axis.

The Riemann zeta function also has an infinite product expansion in $\{\Re s > 1\}$, giving

$$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$

This function also satisfies Riemann's functional equation

$$\zeta(s) = 2^s \pi^{s - 1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1 - s) \zeta(1 - s)$$

where $\Gamma$ is the gamma function.

The Riemann zeta function has so-called trivial zeros at the negative even integers $-2, -4, -6, \dots$, as well as many zeros on the line $\frac{1}{2} + it$. It is conjectured that all the non-trivial zeros of the Riemann zeta function lie on this line, and this is considered to be one of the most important open problems in mathematics.

Reference: Riemann zeta function.

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Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
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What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?" Are there any…
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Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the formula above , but how ? Do there exist other proofs…
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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ The Riemann hypothesis asserts that all the…
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Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log…
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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.
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Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack?

There is a beautiful paper in Physical Review Letters [PRL 118, 130201 (2017), DOI:10.1103/PhysRevLett.118.130201] by Carl Bender, Dorje Brody, and Markus Müller (BBM) on a Hamiltonian approach to the Riemann Hypothesis. The paper is surprisingly…
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Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several $\zeta$ values are connected with $\pi$,…
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Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are excluded: Books by mathematical cranks (especially books by amateurs who claim to…
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A new continued fraction for Apéry's constant, $\zeta(3)$?

As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as $\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$ where the sequence of $u_n$, starting with $n = 1$, is given by…
Tito Piezas III
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Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the evaluation of the zeta function at odd positive…
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Why does the first 100,000 zeroes of the Riemann Zeta function have double-digit sequence count discontinuities at 00,11,22,33,44,55,66,77,88,99?

Why do the first 100,000 zeroes of the Riemann Zeta function have double-digit sequence count discontinuities at 00,11,22,33,44,55,66,77,88,99? I was investigating Benford Law type behaviors and was running some analyses on the web page "Andrew…
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Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and using Parseval's identity, I have computed $\zeta(2)$…
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Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 \rbrace$ and factorize them, we will get…
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