Riemann hypothesis Study Guide

📖 Core Concepts Riemann zeta function \(\zeta(s)\) – originally defined for \(\operatorname{Re}(s)>1\) by the series \(\displaystyle \zeta(s)=\sum{n=1}^{\infty}\frac{1}{n^{s}}\). Euler product – same function equals \(\displaystyle \prod{p\ \text{prime}}\frac{1}{1-p^{-s}}\) for \(\operatorname{Re}(s)>1\). Analytic continuation – via the Dirichlet eta function \(\eta(s)=\sum{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}\) and the relation \(\displaystyle \zeta(s)=\frac{\eta(s)}{1-2^{1-s}}\), extending \(\zeta\) to \(\operatorname{Re}(s)>0\) (except a simple pole at \(s=1\)). Functional equation – the completed function \(\displaystyle \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s)\) satisfies \(\xi(s)=\xi(1-s)\); zeros are symmetric about the line \(\operatorname{Re}(s)=\tfrac12\). Trivial vs. non‑trivial zeros – trivial zeros at negative even integers \(-2,-4,\dots\); all non‑trivial zeros lie in the critical strip \(0<\operatorname{Re}(s)<1\). Riemann Hypothesis (RH) – every non‑trivial zero has real part exactly \(\tfrac12\). Critical line – the vertical line \(\operatorname{Re}(s)=\tfrac12\). Supremum \(\Theta\) – the largest real part of any non‑trivial zero; RH asserts \(\Theta=\tfrac12\). 📌 Must Remember Series definition: \(\displaystyle \zeta(s)=\sum{n=1}^{\infty}\frac{1}{n^{s}}\) (\(\operatorname{Re}(s)>1\)). Euler product: \(\displaystyle \zeta(s)=\prod{p}\frac{1}{1-p^{-s}}\) (\(\operatorname{Re}(s)>1\)). Eta‑relation: \(\displaystyle \zeta(s)=\frac{\eta(s)}{1-2^{1-s}}\). Functional equation: \(\displaystyle \xi(s)=\xi(1-s)\). Trivial zeros: \(s=-2,-4,-6,\dots\). RH statement: \(\displaystyle \forall\rho\;( \zeta(\rho)=0,\;\rho\notin\{-2,-4,\dots\})\Rightarrow \operatorname{Re}(\rho)=\tfrac12\). Prime‑counting error term under RH: \(|\pi(x)-\operatorname{li}(x)| = O\!\bigl(\sqrt{x}\log x\bigr)\). Chebyshev bound: \(|\psi(x)-x| \le \frac{1}{8\pi}\sqrt{x}\log^{2}x\) for \(x\ge599\). Robin’s inequality: \(\sigma(n) < e^{\gamma} n\log\log n\) for all \(n>5040\) ⇔ RH. Li’s criterion: All \(\lambdan>0\) where \(\displaystyle \lambdan=\sum{\rho}\Bigl[1-\bigl(1-\frac1\rho\bigr)^n\Bigr]\). Nyman–Beurling: Density of step‑function span in \(L^2(0,1)\) ⇔ RH. 🔄 Key Processes Extending \(\zeta(s)\) to \(\operatorname{Re}(s)>0\) Compute \(\eta(s)\) via alternating series (converges for \(\operatorname{Re}(s)>0\)). Apply \(\zeta(s)=\eta(s)/(1-2^{1-s})\). Using the functional equation For a given \(s\), evaluate \(\xi(s)\) and replace \(s\) by \(1-s\) to obtain symmetry relations. Zero‑counting (Riemann–von Mangoldt formula) Compute \(N(T)=\frac{T}{2\pi}\log\!\frac{T}{2\pi e}+ \frac78+S(T)+O(1/T)\). Estimate \(S(T)\) (average size \(\approx\sqrt{\log\log T}\)). Verifying RH numerically (Turing’s method) Compute Hardy’s \(Z(t)\) and locate sign changes → zeros on the critical line. Compare counted zeros via \(N(T)\) with sign‑change count. 🔍 Key Comparisons Trivial zeros vs. non‑trivial zeros Location: \(-2,-4,\dots\) vs. \(0<\operatorname{Re}(s)<1\). Cause: poles of the \(\Gamma\)-factor in the functional equation vs. deep analytic structure. RH vs. Generalized RH (GRH) RH: concerns \(\zeta(s)\) only. GRH: same real‑part‑½ claim for all Dirichlet \(L\)-functions. Robin’s inequality vs. Lagarias’s criterion Robin: bound on sum‑of‑divisors \(\sigma(n)\). Lagarias: bound involving harmonic numbers \(Hn\) and \(\exp(Hn)\log Hn\). Both are equivalent to RH but look at different arithmetic functions. ⚠️ Common Misunderstandings “All zeros lie on the critical line” – only the non‑trivial zeros are conjectured to be on \(\operatorname{Re}(s)=\tfrac12\); trivial zeros are off the line. “RH is proved for large heights” – numerical verification up to \(10^{13}\) is impressive but does not constitute a proof. “Zero‑free region at \(\operatorname{Re}(s)=1\) proves RH” – it only shows no zeros on the line \(\operatorname{Re}(s)=1\), not that they must be at \(\tfrac12\). 🧠 Mental Models / Intuition Symmetry picture: Think of \(\xi(s)\) as a mirror that reflects the complex plane across the critical line; any zero off the line forces a partner symmetric about \(\tfrac12\). Prime‑counting error: RH says the “noise” in the distribution of primes is as small as the square‑root of the counting size, like a random walk’s typical deviation. Step‑function density (Nyman–Beurling): Imagine building any square‑integrable function on \((0,1)\) using tiny “stair steps” placed at positions \(1/k\); being able to do this perfectly is equivalent to RH. 🚩 Exceptions & Edge Cases Simple pole at \(s=1\) – \(\zeta(s)\) blows up; not a zero, but essential for analytic continuation. Zeros on the critical line with multiplicity > 1 – RH does not forbid multiple zeros; however, simplicity is conjectured (and follows from stronger statements). Dirichlet eta relation fails at \(s=1\) because denominator \(1-2^{1-s}=0\); this is the pole. 📍 When to Use Which Estimating \(\pi(x)\) – use the RH error term \(O(\sqrt{x}\log x)\) when a sharp bound is needed; otherwise the unconditional PNT error \(O\!\bigl(x\exp(-c\sqrt{\log x})\bigr)\) suffices. Bounding \(\psi(x)\) – apply Schoenfeld’s bound \(\frac{1}{8\pi}\sqrt{x}\log^{2}x\) only under RH. Testing RH numerically – employ Hardy’s \(Z(t)\) and Turing’s method for heights up to the computational limit. Proving equivalent statements – use Li’s \(\lambdan>0\) when working with zero‑sum expressions; use Robin’s inequality for divisor‑function arguments. 👀 Patterns to Recognize Euler product ↔ prime distribution – each factor \((1-p^{-s})^{-1}\) encodes a prime; zeros of \(\zeta\) translate into fluctuations of prime counts. Zero‑density results – statements like “\(o(T\log T)\) zeros with \(\operatorname{Re}(s)>\tfrac12+\varepsilon\)” indicate that most zeros crowd near the critical line. Gram’s law failures – while often one zero per Gram interval, exceptions become common at very high heights; a sudden “bad Gram point” often signals a zero‑pairing anomaly. 🗂️ Exam Traps Confusing trivial zeros with RH statement – answer choices may list “all zeros are at \(-2,-4,\dots\)”; remember RH only concerns non‑trivial zeros. Mis‑applying the functional equation – the equation involves the completed \(\xi(s)\); omitting the \(\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\) factor leads to incorrect symmetry claims. Assuming the zero‑free region near \(\operatorname{Re}(s)=1\) gives the error term – the region yields a weaker error bound; the optimal \(\sqrt{x}\log x\) bound requires the full RH. Equivalence vs. implication – e.g., Robin’s inequality is equivalent to RH, but the statement “RH ⇒ Robin’s inequality” alone is true; exam items may ask which direction is proven. --- All bullets are drawn directly from the provided outline; no external facts were added.