Tail of the zeta function \(\zeta(s)\):

\[\displaystyle\sum_{n\geq N}\dfrac{1}{n^s}=\dfrac{N^{1-s}}{s-1}+\dfrac{1}{2N^s}-s\int_N^\infty\left(\{t\}-\dfrac{1}{2}\right)t^{-s-1}\ \mathrm{d}t\]

#ZetaFunction #RiemannZetaFunction #TailOfFunction #Function #Maths #NumberTheory #Analysis

There is this really awesome series on YouTube about the Riemann-Zeta function which is currently running and seems to want to explain all the mathematic objects related to it properly it would be awesome if you could show the video series some love while it's running. https://youtu.be/4bzSFNCiKrk
#mathematics #mathstodon #primenumbers #riemann #zetafunction

I wrote a bit about the connection between the zeta function ζ(s) and the primes, a relationship discovered by Euler in 1737 (the so-called product formula or as Derbyshire calls it, the "golden key" [1]).

Some of my notes are here: https://davidmeyer.github.io/qc/Euler_product_formula_for_the_Riemann_zeta_function.pdf. The LaTex source is here: https://www.overleaf.com/read/hvqdtdyftqpb.

As always, questions/comments/corrections/* greatly appreciated.

#math #zetafunction #goldenkey #primes #euler

References
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[1] John Derbyshire. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. The National Academies Press, Washington, DC, 2003.

The Riemann hypothesis, explained by Jørgen Veisdal: 🔗 https://www.cantorsparadise.com/the-riemann-hypothesis-explained-fa01c1f75d3f
\[\zeta(s)=0;\ s\notin2\mathbb{Z}^-\implies\Re(s)=\dfrac{1}{2}\]

#RiemannHypothesis #Riemann #Hypothesis #Atiyah #NumberTheory #Conjecture #AnalyticNumberTheory #Mathematics #CantorsParadise #OpenProblem #UnsolvedProblem #PrimeNumber #RiemannZetaFunction #ZetaFunction #NonTrivialZero #LogarithmicIntegral #XiFunction #PrimeNumberTheorem #PrimeNumberDistribution #PrimeCountingFunction #GammaFunction

PRODUCTS OVER PRIME NUMBERS [2/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\left(1+\dfrac{1}{p(p+1)}\right)=\prod_{p\in\mathbb{P}}\dfrac{1-p^{-3}}{1-p^{-2}}=\dfrac{\zeta(2)}{\zeta(3)}=\dfrac{\pi^2}{6\zeta(3)}\]
\[\displaystyle\prod_{p\in\mathbb{P}}\left(1+\dfrac{1}{p(p-1)}\right)=\prod_{p\in\mathbb{P}}\dfrac{1-p^{-6}}{(1-p^{-2})(1-p^{-3})}=\dfrac{\zeta(2)\zeta(3)}{\zeta(6)}=\dfrac{315}{2\pi^4}\zeta(3)\]
#PrimeProducts #RiemannZetaFunction #EulerProduct #ZetaFunction #InfiniteProduct #NumberTheory

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s+1}{p^s-1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s-1}{p^s+1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s+1}{p^s+1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct

https://www.cantorsparadise.com/primes-in-arithmetic-progressions-the-genius-of-dirichlet-203a09b9bfd5

#arithmeticprogressions #dirichlet #primenumbers #euler #zetafunction

Me looking at a formula of :ramanujan: Ramanujan: Yeah, just do partial fractions of the infinite sum, telescope two of the terms, and look at the remaining value of the #ZetaFunction.

Me counting stitches on my #knitting for DPNs: 48 divided by 3 is 18, right?