--- date: 2022-01-26 13:34 modification date: Wednesday 26th January 2022 14:13:23 title: Riemann Zeta aliases: [Riemann-Zeta, zeta, "Zeta function"] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #todo/untagged - Refs: - [The Zeta Book](attachments/zeta_book.pdf) #resources/books - Links: - [[zeta functions]] - [Unsorted/Hasse-Weil L function](Unsorted/Hasse-Weil%20L%20function.md) - [Arithmetic zeta function](Arithmetic%20zeta%20function) - [Dedekind zeta function](Dedekind%20zeta%20function.md) - [L function](L%20function.md) - [Selberg zeta function](Selberg%20zeta%20function) - [motivic zeta function](Unsorted/motivic%20zeta%20function.md) - [Bernoulli numbers](Bernoulli%20numbers) --- - [ ] What are the sum/product definitions of the zeta function? - [ ] Why are the zeros of $\zeta$ confined to $0 < \Re(z) < 1$? - [ ] Why are we interested in the zeros of $\zeta$? # Riemann Zeta - **Definition of $\zeta$**: defined on $A_1 \da \ts{z\in \CC \st \Re(s) > 1}$ as $$\zeta_{\spec \ZZ}(z) \da \sum_{n \in \ZZ_{>0} }n^{-s} = \prod_{p\in \spec \ZZ}(1-p^{-s})\inv = \prod_{\mfm\in \mspec \ZZ} (1 - \Norm(\mfm)^{-s})\inv, $$ where $\Norm(\mfm)\da \size (\ZZ/\mfm)$. - Converges absolutely and uniformly on $A_1$ using the integral test on any open ball strictly contained in $A_1$ - Defines a holomorphic function on $A_1$ since each term$n^{-s} \da e^{-s\log n}$ is holomorphic. - **Completing**: $F(z) \da \pi^{-z} \Gamma(z) \zeta(2z)$ is holomorphic on $A_{1\over 2}$, replace $z$ by $z/2$ to obtain the completed zeta function $$\hat\zeta(z) \da \pi^{-z/2} \Gamma\qty{z\over 2} \zeta(z)$$ which satisfies the functional equation $\hat\zeta(s) = \hat\zeta(1-s)$. - Note $\Gamma(z)$ is nowhere vanishing, meromorphic, with poles only at $\ts{-n\st n\in \ZZ}$, and $\pi^{-z/2}$ is nowhere vanishing and entire. - $\hat \zeta$ has simple poles only at $z=0, 1$, and its only zeros are on $A_0$ and coincide with the zeros of $\zeta(z)$ since $\Gamma(z)$ and $\pi^{-z/2}$ have no zeros on $A_{0}$. - **The critical strip**: the zeros of $\hat\zeta$ and $\zeta$ lie in the critical strip $0 < \Re(z) < 1$, i.e. $\hat \zeta$ is holomorphic and nonvanishing on $A_1$. - This follows because the Euler product for $\zeta(z)$ converges absolutely and uniformly on $A_1$ and has no zeros there. - How to see this: write $$\zeta(z) \cdot P(z) \da \zeta(z)\cdot\prod_{p\in \spec \ZZ}(1-p^{-s}) = 1$$ which holds for all $s \in A_1$. But the RHS is nonzero, and the LHS converges absolutely to a finite number $c(z)\in \CC$. If $\zeta(z) = 0$ for some $z$, then $0 \cdot P(z) = 1$, a contradiction. - Canceling poles: - Insert an additional factor of ${z\choose 2} = {1\over 2}(z(z-1))$ to make $\zeta$ holomorphic on $\CC$. - Adelic interpretation in terms of Gamma factors - One can write $$\zeta(z)= \prod_{p\in \places{\QQ}, \, p < \infty }F_p(z), \qquad \hat\zeta(z) = \prod_{p\in \places{\QQ}}F_p(z) = F_\infty(z)\cdot \zeta(z)$$ so that the contribution from the infinite place is $F_\infty(z) = \pi^{-z/2}\Gamma(z/2)$ and $F_p(z) = (1-p^{-s})\inv$. See [places](Unsorted/Valuations.md); the places $p$ correspond to $\abs{\wait}_p$ and $\abs{\wait}_\infty$ is the usual absolute value on $\QQ$ (the real/archimedean place). - Infinitely many zeros: - Letting $N(T)$ be the number of zeros of $\hat\zeta(s)$ in the critical strip where $0 \leq \Im(z) \leq T$, let $R = \ts{-\eps \leq \Re(z) \leq 1+\eps, 0\leq \Im(z) \leq T}$, use the argument principle and functional equation to derive an asymptotic estimate $$N(T) = {1\over 2\pi i}\int_{\bd R}{\hat\zeta'(z)\over \hat\zeta(z) }\dz \sim {T\over 2\pi} \log\qty{2\over 2\pi \eps}$$. - This immediately yields that $\zeta$ has infinitely many zeros in the critical strip, the hypothesis is they are all on the critical line $1/2$. - Counting zeros on the critical line: define the Hardy $Z\dash$function $$HZ(t) \da e^{i\theta(t)}\zeta\qty{{1\over 2} + t}, \qquad \theta(t) \da\arg\qty{\Gamma\qty{2it + 1 \over 4}} - {\log(\pi)\over 2}t$$ where $\theta(t)$ is the Riemann-Siegel function. - Roughly: use asymptotic expansions of $HZ(t)$ to count its zeros, relate that to $N(T)$ which allows determining if all zeros in $0< \Im(z) < T$ are on the critical line. Thus has been verified by ordering zeros by imaginary parts and showing the first $10^{13}$ are on the critical line. - Special zeta values $\zeta(-n)$ have some kind of $p\dash$adic continuity properties, and can be $p\dash$adically interpolated to give a $p\dash$adic analog of Riemann zeta. - RH is equivalent to the error term in the [[prime number theorem]] being as sharp as possible, i.e. $$\size\ts{p\in \spec \ZZ \st p\leq x} = \int_2^x {1\over \log(t)}\dt + \bigo(x^{1\over 2}\log(x))$$ # Useful identities Useful secondary functions and notions: - Bernoulli numbers: $${x\over e^x-1} = \sum_{n\geq 0} B_n {x^n\over n!},\qquad B_n \in \QQ$$ - Euler shows that for $n\geq 1$ there is an identity $$\zeta(2 n)=(-1)^{n-1} \frac{(2 \pi)^{2 n}}{2(2 n) !} \,B_{2 n}$$ where $B_k = 0$ for $k\geq 3$ odd and the first few values are $$\begin{gathered} B_0=1, \quad B_1=-1 / 2, \quad B_2=1 / 6, \quad B_4=-1 / 30 \\ B_6=1 / 42, \quad B_8=-1 / 30, \quad B_{10}=5 / 66, \quad B_{12}=-691 / 2730 . \end{gathered}$$which yields $$\zeta(2)=\frac{\pi^2}{6}, \quad \zeta(4)=\frac{\pi^4}{90}, \quad \zeta(6)=\frac{\pi^6}{945}, \quad \zeta(8)=\frac{\pi^8}{9450}$$ - The von Mangoldt function $$\Lambda(n) \da \begin{cases}\log p & \text { if } n=p^k \text { for some prime } p \text { and integer } k \geq 1 \\ 0 & \text { otherwise }\end{cases}$$ which satisfies $\log(n) = \sum_{d|n} \Lambda(d)$ and $\log(\zeta(s)) = \sum_{n\geq 2} {\Lambda(n)\over \log(n)}n^{-s}$. - The Mobius function $$\mu(n)= \begin{cases}0 & n \text { not squarefree} \\ (-1)^k & n\text{ squarefree, }k \da \size\ts{\text{prime factors of n}} \end{cases}$$, so $1$ is squarefree and an even number of prime factors, $-1$ if squarefree and odd number of prime factors. - The $\ell$th sum if divisors function $$\sigma_\ell(n) \da \sum_{d| n} d^\ell$$ - The number of divisors function $$d(n) \da \sigma_0(n) \da \sum_{d|n} 1$$ - The Gamma function $$\Gamma(s) \da \int_{\RR_{\geq 0}} e^{-x}x^{s}\, {\dx\over x} \quad = M{\ts e^{-x}}(s)$$ - The second Chebyshev function $$\psi(x) \da \sum_{p^k\leq x}\log(p) = \sum_{n\leq x}\Lambda(n)$$ - Dirichlet convolution: $$(f\convolve g)(n) \da 1\sum_{d|n}f\qty{d}g\qty{n\over d} = \sum_{d_1 d_2=n}f(d_1)g(d_2)$$ - Consequence: $F(s) G(s) = \sum_{n\geq 1} (f\convolve g)(n) n^{-s}$. - The function $\eps(n) \da \floor{1\over n}$ is the convolution algebra unit and $1_\zeta(n) \da 1$ corresponds to $\zeta(s)$. - Mobius inversion: $$g = f\convolve 1_\zeta \iff f = g\convolve \mu$$ - The Melin transform: $$M\ts{f}(s) \da \int_{\RR_\geq 0}x^{s}f(x)\,{\dx\over x} \quad\mapstofrom\quad M\inv\ts{\phi}(x) = {1\over 2\pi i}\int_{c-i\infty}^{c+i\infty} x^{-s}\phi(s)\,\ds$$ - An analog of the Laplace transform for multiplicative functions. Why the weird measure: ${\dx\over x}$ is a multiplicative Haar measure, invariant under dilations $x\mapsto ax$, and $\dx$ is the additive Haar measure which is invariant under translations $x\mapsto a+x$. - Importance: integral formula for zeta function: $$M\ts{1\over e^x-1} = \Gamma(s)\zeta(s)\implies\zeta(s) = {1\over \Gamma(s)}\int_{\RR_{\geq 0}} {x^s:\over e^{x}-1}\, {\dx\over x}$$ - Poisson summation formula: ? Identities: | Dirichlet generating function | Sequence | Explicit equation | |:----------------------------------|:----------------------------------------------------------|:----------------------------------------------------------------------------------------------------------------------------------| | $$F(s)$$ | $f(n)$ | $$F(s) = \sum_{n\geq 1}f(n) n^{-s}$$ | | $$F'(s)$$ | $-f(n)\log(n)$ | $$F'(s) = -\sum_{n\geq 1} f(n)\log(n) n^{-s}$$ | | $$F'(s)\over F(s)$$ | $-f(n)\Lambda(n)$ | $${F'(s)\over F(s)} = -\sum_{n\geq 1} f(n) \Lambda(n) n^{-s}$$ | | $${1\over F(s)}$$ | $\mu(n)f(n)$ | $${1\over F(s)} = \sum_{n\geq 1} \mu(n) f(n) n^{-s}$$ | | $$(F(s))^2$$ | $f(n)\convolve f(n)$ | $$(F(s))^2 = \sum_{n\geq 0} \sum_{d\mid n} f\qty{n}f\qty{n\over d} n^{-s}$$ | | $$\log F(s)$$ | $f(n)\Lambda(n)\over \log(n)$ | $$\log F(s) = \sum_{n\geq 0} {f(n)\Lambda(n)\over \log(n)} n^{-s}$$ | | $$\zeta(s)$$ | $1$ | $$\zeta(s) = \sum_{n\geq 1} 1\cdot n^{-s}$$ | | $$\zeta'(s)$$ | $-1\cdot \log(n)$ | $$\zeta'(s) = -\sum_n \log(n) n^{-s}$$ | | $${\zeta'(s)\over \zeta(s)}$$ | $-\Lambda(n)$ | $${\zeta'(s)\over \zeta(s)} = -\sum_{n\geq 1} \Lambda(n) n^{-s}$$ | | $${1\over \zeta(s)}$$ | $\mu(n)$ | $${1\over \zeta(s)} = \sum_{n} \mu(n) n^{-s}$$ | | $$(\zeta(s))^2$$ | $d(n)$ | $$\zeta^2(s) = \sum_n d(n) n^{-s}$$ | | $$\log\zeta(s)$$ | ${\Lambda(n)\over \log(n)}$ | $$\log \zeta(s) = \sum_{n\geq 2} {\Lambda(n) \over \log(n)} n^{-s}$$ | | $$\zeta(s-1)\over \zeta(s)$$ | $\phi(n)$ | $${\zeta(s-1)\over \zeta(s) } = \sum_{n\geq 1} \phi(n) n^{-s}$$ | # Functional equation - A proof of the functional equation: - ![](attachments/Pasted%20image%2020220430210843.png) - ![](attachments/Pasted%20image%2020220430210919.png) - ![](attachments/Pasted%20image%2020220430210933.png) # Special values ## Rationality - $\zeta(1-2k)$ is rational, the proof follows from showing the Fourier coefficients in the Eisenstein series of [weight](weight) $2k$ and [level](level) 1 $$G_{2 k}(q)=\zeta(1-2 k)+2 \sum_{n \geq 1} \sigma_{2 k-1}(n) q^{n}$$ - Related to [Eisenstein series](Unsorted/Eisenstein%20series.md): $\zeta(z)$ is the constant term of the Eisenstein series $G_{2k}$, and rationality of $\zeta(1-2k)$ follows from properties of $G_{2k}$. ## Zeros ![](attachments/2023-01-11-6.png) # Results - Euler: $\zeta(m) = -{1\over 2}B_m(2\pi i)^m$ where $B_m \cdot m!$ is the $m$th Bernoulli number, i.e. ${x\over e^x-1} = \sum_{m\geq 0}B_m x^m$. Thus $\zeta(2) = \pi^2/6$, which critically uses a product expansion $$\sinh(z) \da {1\over 2}(e^x + e^{-x}) = \pi x \prod_{n\geq 1}\qty{1 + {x^2\over n^2}}$$ - #open/conjectures For $n\geq 3$ odd, is $\zeta(n)$ transcendental? - Considering tensor powers of the [[Carlitz module]] yields formulas for $\zeta_C(m)$ and can be ued to prove transcendence results there. # Random matrices - Relation to [random matrices](Unsorted/random%20matrix.md): ![](attachments/Pasted%20image%2020220430193636.png)