PRH | Educ | 6.4 • A Gentle First Course in Adelic–Quantum Arithmetic

A Gentle First Course in Adelic–Quantum Arithmetic A preparatory, exercise-rich introduction Aleksandar Perišić November 2025 Abstract This booklet offers a gentle, linear-algebra friendly path toward the adelic–quantum (AQ) viewpoint on arithmetic. The slogan is that addition behaves like a wave (shifts), while multiplication behaves like a scale change (dilations and prime steps). These two actions generate the adelic ax+bsymmetry and satisfy the commutation rule [Hidele, Hadele] = i Hadele , which is the adelic avatar of the classical [D, P] = iP on L2(R). We build up the picture from first principles—numbers, primes and composites; vectors and inner products; functions as vectors; Fourier transform; a gentle look at p-adic numbers; then adeles/ideles and their basic actions—with many short exercises (odd-numbered answers included). How to read this booklet Each section ends with bite-size exercises. If an idea feels advanced, we put a quick oneparagraph Aside right next to it. You will need only: high-school algebra, first courses in calculus and linear algebra, and comfort with complex numbers. Measure theory is lightly used and explained when needed. Part I Warm-up Mathematics 1 Numbers, primes, and composites Definition 1.1 (Prime and composite).A number n∈N≥2is prime if its only positive divisors are 1and n. It is composite if n=ab with 1< a < n and 1< b < n. Example 1.2. 2,3,5,7,11 are prime; 9 = 32and 49 = 72are composite. Theorem 1.3 (Fundamental Theorem of Arithmetic).Every n≥2factors uniquely as n= Qpαi iwith pidistinct primes, αi∈N. Exercise Write each number as a product of primes and list its prime exponents:2,9,5,49,360. Then form the vector of exponents (α2, α3, α5, α7, . . . )for each. Which of these vectors is sparse? Why? 1 Aside [Superposition idea] Think of a formal basis {pk:pprime, k ≥0}and attach to an integer n=Qpαpthe superposition |ni=Ppαp|piif you only keep the prime counts. Later we will refine this to include all k≥1with weights. Exercise (Parity and divisibility.) Show nis even iff its prime-exponent vector has α2≥1. How would you recognize divisibility by 9in that vector? 2 Vectors, inner products, and complex amplitudes A vector space is a set where you can add vectors and scale them by numbers. We use complex numbers because phases (angles) matter in waves. Definition 2.1 (Inner product).On a complex vector space V, an inner product h·|·i is linear in the first slot, conjugate-symmetric, and positive definite. Its induced norm is kvk2=phv|vi. Example 2.2 (Column vectors).For u, v ∈Cn,hu|vi=Pjujvj. The matrix of a linear map Tsatisfies hTu|vi=hu|T∗viwhere T∗is the conjugate transpose. Exercise (Unitary matrices.) Show that U∗U=Iiff Upreserves inner products. Give an example of a 2×2unitary that rotates vectors by an angle. 3 Functions as vectors; the Fourier transform Definition 3.1. L2(R)is the space of square-integrable functions with inner product hf|gi= RRf(x)g(x) dx. Proposition 3.2 (Fourier transform).The Fourier transform b f(ξ) = RRf(x)e−2πixξ dxis a unitary map L2(R)→L2(R). Aside [Why Fourier?] Shifts in xbecome phases in frequency. This is why addition (shifts) looks like waves. Exercise (Gaussian.) Show f(x) = e−πx2is its own Fourier transform. 4 A 10-minute view of p-adics Definition 4.1 (p-adic absolute value).For prime p, write n=pkmwith p-m. Define |n|p=p−kand extend to Qby |a/b|p=|a|p/|b|p. Aside [Intuition] |·|pmeasures divisibility by p. Numbers with many p’s in them look small p-adically. 2 Definition 4.2. Qpis the completion of Qunder |·|p. Its integers Zp={x∈Qp:|x|p≤1} form a compact ring. Exercise Compute |9|3,|9|5, and |49|7. Which are ≤1? Which are <1? 5 Adeles and ideles (a friendly first pass) Definition 5.1 (Adeles).The adeles A=Q′ vQvare the restricted product of the completions of Q(real v=∞and all p-adics), where “restricted” means almost all finite components lie in Zp. Definition 5.2 (Ideles).The ideles A×are the units of A, with multiplicative Haar measure d×vand module |v|A=Qv|vv|v. Aside [Takeaway] Adeles let us do addition and Fourier analysis simultaneously over the real line and all p-adic lines. Ideles do the same for multiplication. Exercise (Restricted product.) Explain why the tuple with x∞∈Rand xp∈Zpfor all but finitely many pdefines a point in A. Part II The ax+b story 6 The real ax+b model and [D,P]=iP Let U(a)f(x) = f(x−a)(shift) and (V(λ)f)(x) = λ−1/2f(x/λ)(dilation). Define generators P=−i ∂x, D =1 2(x∂x+∂xx) = −i 2(x P +P x). Then on a natural core (Schwartz functions), [D, P ] = iP. Exercise Verify [D, P] = iP by direct computation on e−πx2. Aside [Meaning] Scaling then shifting is not the same as shifting then scaling; the mismatch is exactly proportional to the shift generator P. 7 From real to adelic: the Hilbert space H Let H=L2(A/Q,dx), the space of square-integrable functions on adeles modulo rationals. The adelic affine group G=AoA×acts by (U(a, v)f)(x) = |v|−1/2 Afv−1(x−a),(a, v)∈A×A×.(1) 3 Definition 7.1 (Reference state).Take Φ = NvΦvwith Φ∞(x) = π−1/4e−πx2and Φp=1Zp. Periodize to ΘΦ(x) = Pq∈QΦ(x+q)and normalize it in H. Exercise (Orthogonality under shifts.) Show hΘΦ|U(a, 1)ΘΦiis 1if a∈Qand decays rapidly otherwise. 8 Two flows: additive and idelic Additive flow. U(t, 1) with generator Hadele =id dtt=0 U(t, 1). At v=∞this is the usual momentum −i∂x∞; at finite places it is the infinitesimal version of x7→ x+t. Idelic flow. Split A×into the connected norm flow and prime steps. Pick vu∈A×with |vu|A=euand set N=id duu=0 U(0, vu). For each prime p, let Tp=U(0, ϖp)multiply by the idèle that is pat place pand 1elsewhere. Formally write Tp=ei(log p)Npto define counting operators Npon the prime band. Define Hidele := N+X p (log p)Np. Aside [Domains] All commutators are taken on the common invariant core of Schwartz–Bruhat vectors. Stone’s theorem ensures self-adjoint generators of unitary flows. Theorem 8.1 (Adelic ax+bcommutator).On the core, [Hidele, Hadele] = i Hadele. Exercise (Real place check.) Show that at the real place alone, [D, P] = iP matches the N–Hadele part of the relation. Exercise (Prime steps as phases.) Show Tpacts by a phase on Fourier modes supported on multiples of log pin the frequency of A. 9 A third flow: prime–phase gauge (Helson twist) Besides shifting (additive flow) and scaling/prime steps (idelic flow), there is a third, commuting unitary flow that only rephases the prime band. Fix real phases {θp}pprime and define V(t) = expi t Hgauge, Hgauge := X p θpNp, t ∈R, 4 so that on the tooth u=klog pone has V(t) : δu−klog p7−→ ei t k θpδu−klog p. Equivalently, at the Euler–product level this is the Helson twist by a completely multiplicative unimodular function χt(p) = eitθp, χt(n) = Y pk∥n χt(p)k, ζχt(s) = Y p 1 1−χt(p)p−s=X n≥1 χt(n) ns. Proposition 9.1 (Commutation with the two flows).On the common Schwartz–Bruhat core, [Hgauge, Hadele] = 0,[Hgauge, Hidele] = 0. Idea. Npact only on the discrete prime band, hence commute with additive shifts. They also commute with the norm generator Nand with each other, so Hgauge commutes with Hidele =N+Pq(log q)Nq. Aside [Why phases?] Keeping |χt(p)|= 1 preserves absolute convergence in Re s > 1and complete multiplicativity of Dirichlet coefficients. The gauge flow rephases prime powers but does not translate them (additive) or rescale them (multiplicative). Moreover, in the Helson class one can prescribe zero/pole sets in the half-plane Re s < 1—in particular, realize models whose nontrivial zeros all lie on Re s=1 2—by a suitable choice of phases at the primes [7, 6, 5, 4]. Exercise (Semigroup law.) Show that for the gauge flow V(t) = exp(i t Hgauge)one has V(t+s) = V(t)V(s)and V(0) = I. 10 A fourth flow: torus/Floquet closure Pick a length L > 0and wrap the scale variable u∈Rto the circle θu:= umod L∈S1 u∼ =R/LZ. To “close the geometry” while keeping full control, introduce a Floquet (twist) parameter α∈ R/2πZand the Floquet periodization operator (ΠL,αf)(θu) := X n∈Z einα f(θu+nL). This identifies u∼u+Lwith quasi–periodic boundary condition f(u+L) = eiαf(u). The corresponding torus flow is the 1–parameter unitary group W(α) := eiα w,so that ΠL,α =W(α) ΠL,0, where wis the (integer) winding operator recording how many wraps by Lwe shift in the u–direction. Proposition 10.1 (Commutation with the three flows).On the Schwartz–Bruhat core, [W(α), Hadele ] = 0,[W(α), Hidele ] = 0,[W(α), Hgauge ] = 0. 5 Idea. Hadele and the connected part of Hidele act by translations in u, which commute with periodization ΠL,α. The discrete prime counters Np(hence Hgauge) act only on the prime band and are independent of u–wrapping, so they commute with W(α)as well. Aside [Geometry in one line] Hadele slides/blurs along the base circle S1 u; Hidele reweights by u(or performs the heat time change) before periodization; Gauge spins the phase circle at each prime tooth. The torus flow W(α)merely fixes the “gluing rule” u∼u+Lvia the phase eiα. Remark 10.2 (Relation to horizontal tilt).On the torus, a real horizontal tilt edu corresponds to a nonunitary holonomy edL. To keep unitarity in this fourth flow, use α∈R(unit modulus). If you need the real tilt, work on the cylinder (u∈R) or treat it as a line-bundle twist with monodromy edL. 11 Prime band and projectors Let Ω = {klog p:pprime, k ∈N}be the “prime band” in the multiplicative spectrum. One can define a (formal) projector ΠΩthat isolates these frequencies. Exercise (Detecting composites.) Argue why ΠΩannihilates contributions from prime-free integers and explain how pkappear with weight k. 12 A one-vector picture (the simplest story) Aside State: a single vector |ψi ∈ H =L2(A/Q). Additive action (wave): |ψi 7→ U(t, 1) |ψi; generator Hadele (like momentum). Multiplicative action (prime/scale): |ψi 7→ U(0, v)|ψi; generator Hidele (norm flow N+ prime counters Np). Key relation: doing a tiny scale then a tiny shift differs from shift then scale by exactly itimes the shift: [Hidele, Hadele] = iHadele. Example 12.1 (Toy superposition).Consider the superposition |χi=|2i+|9i+|5i+|49i. The counting operators satisfy N2|χi=|2i,N3|χi= 2 |9i,N5|χi=|5i,N7|χi= 2 |49i, others 0. Thus Hidele |χi= (log 2) |2i+ 2(log 3) |9i+ (log 5) |5i+ 2(log 7) |49i+N|χi. Exercise (Weights.) Generalize the example to |χi=Pn≤100 |niand compute Pp(log p)Np|χi explicitly in terms of prime powers ≤100. 6 Part III Arithmetic readouts (ideas, not proofs) 13 Poisson summation and a quantum Weil identity Periodizing the reference vector (forming a theta function) and using Poisson summation yields an identity matching the classical Weil explicit formula, but here it appears before any factorization into primes is performed. Aside [Very short dictionary] Theta/Poisson ↔spectrum matching; zeta zeros ↔resonances; archimedean factor ↔Gaussian envelope. Reference state and additive operators. Let H=L2(A/Q, dx)with the additive unitary U(a, 1) from (1). Take the factorable reference vector Φ = O v Φv,Φ∞(x) = π−1/4e−πx2,Φp=1Zp, and periodize it to the theta state ΘΦ(x) := X q∈Q Φ(x+q)∈ H,kΘΦk2= 1 (after normalization). For a Schwartz–Bruhat test f∈ S(A)define the (additive) Weyl/Heisenberg observable (Wfψ)(x) := ZA f(y)ψ(x−y)dy (strong integral on H), and let Fbe the global additive Fourier transform on L2(A/Q), unitary with kernel e−2πi⟨x,ξ⟩ built from the standard local characters. Aside [Fourier/Haar conventions] The global additive Fourier transform Fis taken with respect to the standard self-dual character ψ(x) = ψ∞(x∞)Y p ψp(xp), ψ∞(t) = e−2πit, ψp(x) = e2πi {x}p, where {x}p∈Q/Zis the p–adic fractional part. Haar measures are chosen self-dual: dx∞=Lebesgue on R(so \ e−πx2=e−πξ2), dxpwith vol(Zp) = 1 and b 1Zp=1Zp. With these choices, the global measure dx =Qvdxvmakes Funitary on L2(A/Q)and the adelic Poisson summation holds without extra constants. Theorem 13.1 (Quantum Weil identity).For the normalized theta state ΘΦand every f∈ S(A), the adelic expectations µΦ(f) := hΘΦ|WfΘΦi, µ∨ Φ(f) := ΘΦF−1WfFΘΦ coincide: µΦ(f) = µ∨ Φ(f). This is a single adelic identity (a quantum Poisson/Weil statement), holding before any placewise factorization. 7 Aside [Operators in one line] Wfaverages additive translations against f; conjugation by F swaps translation with modulation. The theorem asserts the equality of the two expectation values in the theta state ΘΦ. Gaussian readout (baseline) To extract arithmetic data along the scale line, push forward a real test on u= log |v|A: take ft(u) = e−u2/(4t), gt(s) = ZR ft(u)e(s−1 2)udu =√4πt et(s−1 2)2. The quantum identity yields the classical-looking readout X ρ gt(ρ) = X pX k≥1 Λ(pk)ft(klog p) + A∞(t) + E(t), E(t) = 2√4πt et/4.(2) Readout A: Hadele + Hidele only Two commuting deformations: •Hadele (additive blur). Convolution in uby Kσ(u) = 1 √2πσ e−u2/(2σ2)sends t7→ t+σ2/2. •Hidele (horizontal blur). The horizontal heat eτ2 2∂2 ssends t7→ teff =t 1−2τ2t(for 0<t< 1/(2τ2)). Hence X ρ eteff(ρ−1 2)2=X p,k Λ(pk)e−teff (klog p)2+A∞(teff) + 2√4πteff eteff /4. Readout B: adding the gauge flow (third flow) Let χt(p) = eitθp(completely multiplicative, |χt(p)|= 1). Replace {ρ}by zeros {ρt}of Ft(s) = Qp(1 −χt(p)p−s)−1. Then X ρt eteff (ρt−1 2)2=X p,k Λ(pk)eitkθp |{z} gauge phase e−teff (klog p)2+A∞(teff) + 2√4πteff eteff /4. Readout C: all four flows (with torus/Floquet closure) Fix L > 0and a Floquet parameter α∈R/2πZ. Periodize along the u–circle with quasi–period eiα: K(L,α) t(θ) = X n∈Z exp−(θ+nL)2 4teinα, θ =θu−θu(k, p), and include optional strengths rp∈(0,1] so rpk=rk p. Then X ρt eteff (ρt−1 2)2=X p,k Λ(pk)rk p |{z} strength eitkθp |{z} gauge K(L,α) teff θu−(klog p) mod L | {z } torus closure +A∞(teff) + 2√4πteff eteff /4. 8 Exercise (Real Poisson.) Prove the one-dimensional Poisson summation formula on Rfor Schwartz functions and test it on the Gaussian. 14 Gaussian probes and the RH signal Phase-scanned Gaussian test vectors can convert the presence of any off-line zero of ζ(s)into exponential growth on the prime side. Operationally, one seeks growth-detection statistics whose boundedness is equivalent to all nontrivial zeros lying at Re(s) = 1 2. Aside [Caution] Turning these ideas into proofs requires careful operator domains, spectral decompositions, and trace-class manipulations. Here we only sketch the mechanisms. Exercise (Saddle-point feel.) On L2(R), show how a Gaussian window localizes frequency and how a small phase tilt shifts its center. Interpret in terms of detecting a nearby resonance. A derivative-based RH probe (worked example, m= 1) Recall the Gaussian pair ft(u) = e−u2/(4t), gt(s) = √4πt et(s−1 2)2, and the explicit formula readout X ρ gt(ρ) = X pX k≥1 (log p)e−t(klog p)2 | {z } P(t) +A∞(t) + E(t), E(t) = 2√4πt et/4. Define the Chebyshev heat moments H2m(t) := X pX k≥1 (log p) (klog p)2me−t(klog p)2(m≥0), so that P(m)(t) = (−1)mH2m(t). Proposition 14.1 (RH ⇒a first derivative inequality).Set Z(t) := Pρet(ρ−1 2)2. Under RH, (−1)mZ(m)(t)≥0for all m≥0, t > 0. In particular, for m= 1, H2(t)≥A′ ∞(t) + E′(t) (t > 0). Sketch. Differentiate the readout once: Z′(t) = −H2(t) + A′ ∞(t) + E′(t). Under RH, every zero is s=1 2+iγ and Z′(t) = Pγγ2e−tγ2≥0with a global minus sign, i.e. Z′(t)≤0. Hence H2(t)≥A′ ∞(t) + E′(t). 9 Exercise Ex.26 (Infinitesimal dilation). Let (Vλf)(x) = λ−1/2f(x/λ)for λ > 0. Show the generator at λ=euis N=−i(x∂x+1 2). Solution Set u= log λ. Then Veuf(x) = e−u/2f(e−ux). Differentiate at u= 0:d du |0Veuf= −(1 2f+xf′). Thus N=id du |0Veu=−i(x∂x+1 2). Exercise Ex.27 (Commutator with momentum). With P=−i∂xand Nfrom Ex.26, verify [N, P] = iP. Solution Use [x∂x, P] = [x, −i∂x]∂x+x[∂x, P] = i∂x. Also [1 2, P] = 0. Hence [N, P] = −i[i∂x] = iP . Exercise Ex.28 (Prime counters in a toy basis). On the formal basis {pk}, define Nppk= kpkand Nqpk= 0 for q6=p. Compute Hidele|2i+|9i+|5i+|49iignoring the smooth N-part. Solution (log 2) |2i+ 2(log 3) |9i+ (log 5) |5i+ 2(log 7) |49i. Exercise Ex.29 (Poisson for a Gaussian family). Show X n∈Z e−πtn2=t−1/2X m∈Z e−πm2/t for t > 0. Solution Apply Poisson summation to ft(x) = e−πtx2, whose Fourier transform is b ft(ξ) = t−1/2e−πξ2/t. Then Pnft(n) = Pmb ft(m). Exercise Ex.30 (Fourier tilt/shift). Show Fe2πiaxf(x)(ξ) = b f(ξ−a). Solution Rf(x)e2πiaxe−2πixξ dx =Rf(x)e−2πix(ξ−a)dx =b f(ξ−a). Exercise Ex.31 (Unitary and generator.) Show V(t)is unitary and that id dtt=0V(t) = Hgauge using the spectral rule Nppk=kpk. 16 Solution On the prime-power basis {pk}we have Hgauge pk=θpkpkand hence V(t)pk= eit θpkpkwith unimodular factor eit θpk. Therefore kV(t)pkk=kpkkand, by linearity/density, V(t)is unitary. Differentiating V(t) = eitHgauge at t= 0 gives d dt t=0V(t) = i Hgauge, i.e. id dt t=0V(t) = Hgauge. Exercise Ex.32 (Commutators vanish.) Verify directly on the prime-power basis that [Hgauge,Nq] = 0 for all q, and deduce [Hgauge, Hidele] = 0. Why does [Hgauge, Hadele] = 0 hold at every place? Solution Since the number operators commute, [Np,Nq] = 0 for all p, q, we get [Hgauge,Nq] = PpθpNp,Nq= 0. Thus [Hgauge, Hidele] = [Hgauge, N +X q (log q)Nq] = [Hgauge, N] + X q (log q)[Hgauge,Nq] = [Hgauge, N]. But Ngenerates the connected norm flow and commutes with each prime step Tp= ei(log p)Np, so [N, Np] = 0 and hence [Hgauge, N] = 0. Therefore [Hgauge, Hidele] = 0. Finally, Hadele acts by additive translations (differentiation in the additive variable), while Hgauge acts diagonally on the discrete prime band; they operate on independent degrees, so [Hgauge, Hadele] = 0 on the Schwartz–Bruhat core. Exercise Ex.33 (From phases to Euler factors.) Starting from V(t)acting by eitkθpon the tooth klog p, derive that Qp(1 −χt(p)p−s)−1has Dirichlet coefficients χt(n)with χt completely multiplicative and |χt(n)|= 1. Solution On each tooth klog p,V(t)contributes a phase eitkθp, i.e. in Euler-factor form 1 1−χt(p)p−s=X k≥0 χt(p)kp−ks, χt(p) := eitθp. Multiplying over primes and expanding, the coefficient of n=Qpkpis Qpχt(p)kp= χt(n), which is completely multiplicative with |χt(n)|= 1. Thus Qp(1 −χt(p)p−s)−1= Pn≥1χt(n)n−sas claimed. Exercise Ex.34 (Tiny finite example.) Fix θ2=π/3,θ3=−π/4and θp= 0 for p≥5. Compute the twist factors on 2,4,8,3,9,6,12 at time t= 1 (remember 6 = 2 ·3gets χ(2)χ(3)). 17 Solution With θ2=π/3,θ3=−π/4,t= 1: χ(2) = eiπ/3=1 2+i√3 2, χ(3) = e−iπ/4=1 √2(1 −i). Then 2 : χ(2) = eiπ/3,4 = 22:χ(2)2=ei2π/3,8 = 23:χ(2)3=eiπ =−1, 3 : χ(3) = e−iπ/4,9 = 32:χ(3)2=e−iπ/2=−i, 6 = 2 ·3 : χ(2)χ(3) = ei(π/3−π/4) =eiπ/12, 12 = 22·3 : χ(2)2χ(3) = ei(2π/3−π/4) =ei5π/12. (Optionally: eiπ/12 = cos(15◦) + isin(15◦)and ei5π/12 = cos(75◦) + isin(75◦).) Exercise Ex.35 (Twisted Gaussian identity.) Assume the untwisted Gaussian explicit formula. Insert the factor eitkθpon the prime side and explain why the same identity holds with the zero set replaced by {ρt}(zeros of ζχt). Solution Start from the untwisted explicit formula with Gaussian probe ft0: X ρ gt0(ρ) = X p,k Λ(pk)ft0(klog p) + A∞(t0) + E(t0). Replacing the Euler factors by (1 −χt(p)p−s)−1multiplies the prime-power contributions by χt(p)k, so the prime side becomes Pp,k Λ(pk)χt(p)kft0(klog p). By the same contour/Poisson argument (Weil’s explicit formula applied to ζχt), the left-hand side is the sum over zeros of ζχt, i.e. Pρtgt0(ρt), while the archimedean and endpoint terms are unchanged (the twist is at finite primes only). Hence the same identity holds with the zero set replaced by {ρt}. Exercise Ex.36 (First motion of a zero under a gauge twist). Let χt(p) = eitθpand Ft(s) = ζχt(s) = Qp(1 −χt(p)p−s)−1. Assume ρis a simple zero of ζ(so ζ(ρ) = 0 and ζ′(ρ)6= 0), and let ρtbe the zero of Ftwith ρ0=ρ. Show that ρ′(0) = 0 and compute ρ′′(0) = −L1(ρ)2+L2(ρ), L1(s) = iX p θpp−s 1−p−s, L2(s) = −X p θ2 pp−s (1 −p−s)2. (Hint: expand Ft(s)from log Ft(s) = log ζ(s) + tL1(s) + t2 2L2(s) + O(t3)and use the implicit function theorem on Ft(ρt) = 0.) 18 Solution Write H(s, t) := Ft(s)and enforce H(ρt, t)=0with ρt=ρ+δ(t),δ(0) = 0. From log Ft(s) = log ζ(s) + tL1(s) + t2 2L2(s) + O(t3)we get Ft(s) = ζ(s)1 + tL1(s) + t2 2L1(s)2+L2(s)+O(t3). Taylor-expand in sat s=ρ: ζ(ρ+δ) = ζ′(ρ)δ+1 2ζ′′(ρ)δ2+O(δ3), Lj(ρ+δ) = Lj(ρ) + O(δ). Keep terms up to t2and note δ=ρ′(0)t+1 2ρ′′(0)t2+O(t3). Then 0 = Ft(ρ+δ) = ζ′(ρ)δ+O(δ2)1 + tL1(ρ) + t2 2L1(ρ)2+L2(ρ)+O(t3). Collect coefficients by powers of t: - At order t:ζ′(ρ)ρ′(0) = 0 ⇒ρ′(0) = 0 (since ζ′(ρ)6= 0). - At order t2: using ρ′(0) = 0, we have 0 = ζ′(ρ)1 2ρ′′(0) + 0 ·L1(ρ)+0 ⇒ρ′′(0) = −L1(ρ)2+L2(ρ). (The last equality comes from inserting the t2expansion factor t2 2L2 1+L2multiplying ζ′(ρ)δand matching coefficients.) Thus ρ′(0) = 0 and ρ′′(0) = −L1(ρ)2+L2(ρ). Exercise Ex. 37 (Floquet periodization and quasi–periodicity). Fix L > 0and α∈R. Define the Floquet periodization (ΠL,αf)(θ) := X n∈Z einα f(θ+nL). (a) Show that g(θ) := (ΠL,αf)(θ)satisfies the quasi–periodic boundary condition g(θ+ L) = eiαg(θ). (b) Conversely, if gis defined on [0, L)and obeys g(θ+L) = eiαg(θ)(by extension), show g= ΠL,αfwith f:= g1[0,L). Solution Ex. 37. (a) Compute g(θ+L) = X n∈Z einα f(θ+L+nL) = X m∈Z ei(m−1)αf(θ+mL) = eiα g(θ). (b) For θ∈[0, L)we have (ΠL,αf)(θ) = f(θ) = g(θ). For general θ, write θ=θ0+nL with θ0∈[0, L); then (ΠL,αf)(θ) = einαf(θ0) = einαg(θ0) = g(θ), using the quasi–periodicity of g. Hence g= ΠL,αf. 19 Exercise Ex. 38 (Fourier series of the periodized Gaussian kernel). For t > 0and L > 0, define the Floquet heat kernel on the u–circle K(L,α) t(θ) := X n∈Z exp−(θ+nL)2 4teinα. Show that K(L,α) tadmits the Fourier series K(L,α) t(θ) = 1 LX m∈Z exp−t2πm +α L2expi2πm +α Lθ. (Hint: Apply Poisson summation to x7→ e−x2/(4t)eiαx/L evaluated on the lattice x= θ+nL.) Solution Ex. 38. Consider F(x) = e−x2/(4t)eiαx/L on R. Poisson summation gives X n∈Z F(θ+nL) = 1 LX m∈Zb F2πm Lei2πm Lθ. A direct Gaussian transform yields b F(ξ) = √4πt exp−t(ξ−α/L)2. Plugging ξ=2πm L gives X n e−(θ+nL)2/(4t)einα =1 LX m∈Z exp−t2πm +α L2ei2πm+α Lθ, which is the claimed series. Exercise Ex. 39 (Semigroup and heat equation on the u–circle). Let ∗denote convolution on the circle of length L:(f∗g)(θ) = 1 LRL 0f(θ−η)g(η)dη. Show that K(L,α) t1∗K(L,α) t2=K(L,α) t1+t2(t1, t2>0), and that for any quasi–periodic g0with g0(θ+L) = eiαg0(θ), the function g(θ, t) = (K(L,α) t∗g0)(θ)solves ∂tg=∂2 θgon the circle with the same quasi–periodic boundary condition. 20 Solution Ex. 39. Using the Fourier series from Ex. 38, K(L,α) t(θ) = 1 LX m∈Z e−tλ2 meiλmθ, λm:= 2πm +α L. Then convolution multiplies Fourier coefficients, so \ Kt1∗Kt2(m) = d Kt1(m)d Kt2(m) = e−t1λ2 me−t2λ2 m=e−(t1+t2)λ2 m, which equals \ Kt1+t2(m). For the heat equation, ∂tg=X m (−λ2 m)e−tλ2 mbg0(m)eiλmθ=∂2 θX m e−tλ2 mbg0(m)eiλmθ=∂2 θg. Each mode eiλmθobeys eiλm(θ+L)=eiαeiλmθ, hence g(θ, t)is quasi–periodic with parameter α. Exercise Ex. 40 (Fourth flow commutes with the other three). Let W(α)be the torus/Floquet closure operator implementing u∼u+Lwith phase eiα (equivalently, acting via ΠL,α on test functions). Show on the common core that [W(α), Hadele] = [W(α), Hidele] = [W(α), Hgauge] = 0. (Hint: Hadele generates translations in u;Hidele splits into a connected norm flow in uand prime counters independent of u;Hgauge acts only on the prime band.) Solution Ex. 40. Translations in ucommute with periodization and with the quasi–periodic boundary condition, hence [W(α), Hadele] = 0. The connected part of Hidele is a (logarithmic) dilation acting as a translation in uafter taking logs, so it also commutes with W(α); the discrete prime counters Npdo not touch the u–variable, hence [W(α),Pp(log p)Np] = 0, giving [W(α), Hidele] = 0. Finally, Hgauge =PpθpNpacts only on the prime band, independent of u, so [W(α), Hgauge] = 0. Exercise Ex. 41 (Aliasing on the torus: a concrete computation). Take L= log 8. Compute the positions on the u–circle θu(k, p) := (klog p) mod Lfor (p, k)∈ {(2,1),(2,2),(2,3),(3,1)}. With phases θ2=π/3,θ3=−π/4,t= 1, and strengths r2=r3= 1, list the four torus points (θu, ϕ)where ϕ=t k θpmod 2π. Which teeth alias at the same θu? 21 Solution Ex. 41. Since L= log 8 = 3 log 2: θu(2,1) = log 2, θu(2,2) = 2 log 2, θu(2,3) = 3 log 2 ≡0 (modL). For p= 3,θu(1,3) = log 3 which is not a multiple of log 2, so it is distinct mod L. Phases at t= 1: ϕ(2,1) = π 3, ϕ(2,2) = 2π 3, ϕ(2,3) = π, ϕ(3,1) = −π 4(mod 2π). Thus the four torus points are (log 2, π/3),(2 log 2,2π/3),(0, π),(log 3 mod log 8,−π/4). Aliasing occurs because 3 log 2 ≡0 ( mod L), so the tooth (p, k) = (2,3) lands at the same θuas (p, k)=(any,0) (i.e. the origin of the u–circle). The other three positions are distinct. Further reading and references References [1] J. T. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, Ph.D. thesis, Princeton, 1950; reprinted in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Academic Press, 1967, pp. 305–347. [2] A. Weil, Basic Number Theory, 2nd ed., Springer, 1974. [3] G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. (For physics-style analogies: the ax+bgroup and its unitary representations.) [4] H. Helson, Compact groups and Dirichlet series,Ark. Mat. 8(1969), 139–143. (Classical starting point for Dirichlet series with unimodular completely multiplicative coefficients.) [5] K. Seip, Universality and distribution of zeros and poles of some zeta functions,J. Anal. Math. 141 (2020), 331–381; arXiv:1812.11729. (Foundational analysis of Helson zeta zeros/poles and universality.) [6] I. Bochkov and R. Romanov, On zeroes and poles of Helson zeta functions, arXiv:2106.15949, 2021. (Arbitrary zeros/poles in 21/40 <Re s < 1unconditionally; in 1/2<Re s < 1under RH.) [7] J. Andersson, Mittag–Leffler type theorems for Helson zeta–functions, arXiv:2408.15713, 2024. (Meromorphic continuation with prescribed zeros/poles in Re s < 1for Helson zetas.) 22