HBP | Question | 4.1 • Primes Revisited: Hopf-Algebraic Origins

Primes Revisited: Hopf-Algebraic Origins Aleksandar Perišić August 2025 Abstract Traditionally one writes the Euler product almost as an axiom: ζ(s) = ∏ pprime (1−p−s)−1. Using minimal possible theoretical settings, we show that the Euler product—and thus the notion of “prime” as a multiplicative irreducible—emerges from the Dirichlet coalgebra framework, where the coproduct is not an algebra morphism; consequently log ζis not primitive. Working in the half–plane of absolute convergence, we cast Dirichlet series via value fibers: for D(s) = ∑ n≥1 a(n)n−swith Ey(s):=∑ n≥1 a(n)=y n−s, we study the exponentialized transform exp ∑ y y Ey(s)=∏ b∈B F(b−s),F(z) = exp ∑ k≥1 f(k)zk. We prove a simple tower–invariance criterion (Theorem 1.1) that is necessary and sufficient for such an Euler–product decomposition, and give a greedy slope–peeling algorithm—with explicit finite–difference error bounds—to reconstruct the tower primitive base set Band the weights f(k)from a single fiber. Applied to log ζ, this yields F(z) = (1−z)−1and finally recovers primes without assuming unique factorization. We also present concrete non–examples that diagnose failure when tower–invariance does not hold. Hopf-Algebraic Exponential Formula Let Hbe a connected graded Hopf algebra over C. An element p∈H,∆(p) = p⊗1+1⊗p is primitive, and g∈Hwith ∆(g) = g⊗gis group-like. In a complete connected Hopf algebra, exp : Prim(H)←→ G(H): log, i.e. exp(p) = ∑n≥0pn/n!is group-like, and conversely log(g)is primitive. Primitive Bases from the Dirichlet Coalgebra Take the Dirichlet coalgebra H=M n≥1 Cn−s,b H=∏ n≥1 Cn−s,∆(n−s) = ∑ d|n d−s⊗(n/d)−s. Then ζ(s) = ∑n≥1n−sis group-like in b H: ∆ζ(s) = ζ(s)⊗ζ(s). 1 Remark (Coalgebra primitives and multiplicative irreducibles).An element n−sis coalgebra–primitive if ∆(n−s) = n−s⊗1+1⊗n−s. From ∆(n−s) = ∑ d|n d−s⊗(n/d)−s=n−s⊗1+∑ 1<d<n+1⊗n−s, we see that n−sis coalgebra–primitive precisely when there are no intermediate divisors 1< d<n; equivalently, nis a multiplicative irreducible in (N,×). We will refer to such nas primitive bases. This identification is all we need at this stage: it singles out the “atoms’’ for the coproduct. In later sections, without assuming unique factorization, we will recover that these primitive bases index the local factors. Let Bdenote the set of primitive bases (for now, only in tower-base sense explained later). For each (tower) primitive base p∈B, which is also coalgebra–primitive, set gp:= (1−p−s)−1=∑ k≥0 p−ks ∈b H. Because the divisors of pkare piwith 0≤i≤k, ∆(gp) = ∑ k≥0 k ∑ i=0 p−is ⊗p−(k−i)s=∑ i≥0 p−is⊗∑ j≥0 p−js=gp⊗gp, so gpis group-like. Hence, as an identity in b H(and analytically for ℜs>1), ζ(s) = ∏ p∈B gp=∏ p∈B (1−p−s)−1. The whole point of the construction is that, by the end, the qualifier “tower” can be safely dropped: the tower–primitive bases recovered from the pipeline will coincide with the coalgebra–primitive elements (the actual primes for ζ). For now, the equality between the sum and product forms of ζis to be understood purely as a formal identity in b H. + If log ζwere primitive for this ∆, we would have nothing to do: the two desired sides would coincide out-of-the-box. But it is not. Remark (Why log ζis not primitive for this ∆).Linearity yields ∆(log ζ) = ∑ p∈B ∑ k≥1 1 k k ∑ i=0 p−is ⊗p−(k−i)s, which includes mixed terms (e.g. 1 2p−s⊗p−s) absent from (log ζ)⊗1+1⊗(log ζ). The usual statement “log(group-like)is primitive’’ requires ∆to be an algebra morphism, which the divisor-sum coproduct is not. Remark (Duality with Dirichlet convolution).Writing en:=n−s, the coproduct ∆(en) = ∑d|ned⊗en/dis dual to Dirichlet convolution (f∗g)(n) = ∑d|nf(d)g(n/d). Assumptions and Standing Conventions 1. Absolute convergence. There exists σ0such that ∑n≥1|a(n)|n−σ<∞for all σ>σ0. All termwise manipulations (rearrangement, log/exp of Dirichlet series) occur in such a half–plane. We write σ=ℜs. 2 2. Completion. Infinite Dirichlet series live in the completed space; when a coalgebra/Hopf structure is present we use the completion b H(product topology) so that exponentials/logarithms of absolutely convergent series are defined. 3. Tower primitive normalization. Every n>1can be written uniquely as n=bkwith b not a perfect power (the tower primitive base). All uniqueness statements for factorizations across towers are understood after this normalization. Value–Fiber Decomposition Let Spec(a):={a(n):n∈N}(automatically countable). Define, in any half–plane of absolute convergence, Ey(s):=∑ n≥1 a(n)=y n−s,D(s):=∑ y∈Spec(a) y Ey(s) = ∑ n≥1 a(n)n−s, and the exponentialized Dirichlet transform Prod[D](s):=exp∑ y∈Spec(a) y Ey(s)=exp∑ n≥1 a(n)n−s. Exponential Map and the Uniform Local Factor Theorem 1.1 (Exponential Map Criterion).Fix the tower primitive normalization (every n=bk with bnot a perfect power). The following are equivalent in any half–plane of absolute convergence. (i) There exists a set B⊂ {2, 3, . . . }of tower primitive bases and a sequence f:N≥1→Csuch that Prod[D](s) = ∏ b∈B F(b−s),F(z) = exp∑ k≥1 f(k)zk. (ii) (Tower–invariance) For the tower primitive representation n=bkone has a(n) = (f(k),b∈B, 0, b/∈B. Moreover, letting the level fibers E(k)(s):=∑b∈Bb−ks, we have D(s) = ∑ k≥1 f(k)E(k)(s),Ey(s) = ∑ {k:f(k)=y} E(k)(s). After tower primitive normalization the pair (B,f)is unique. Proof. In a half–plane of absolute convergence, log Prod[D](s) = ∑ y yEy(s) = ∑ n≥1 a(n)n−s. If (i) holds, then log Prod[D](s) = ∑ b∈B ∑ k≥1 f(k)b−ks. By tower primitive normalization each nhas at most one representation n=bkwith b∈B, hence uniqueness of Dirichlet coefficients yields (ii). Conversely, (ii) implies the last display, and exponentiation gives (i). The formulas for E(k)and Eyare immediate. 3 Recovering the Bases from a Single Fiber Assume Theorem 1.1. Work with the level fibers E(k)(s):=∑ b∈B b−ks,k≥1. For k=1, set S(σ):=E(1)(σ) = ∑ b∈B b−σ,b1<b2<· · · . Lemma 1.2 (First base from the slope). log b1=lim σ→∞−∂σlog S(σ),with −∂σlog S(σ) = log b1+O(b1/b2)σ. Proof. Factor b−σ 1: S(σ) = b−σ 11+∑ j≥2 (b1/bj)σ, then differentiate log S; the error is O((b1/b2)σ). Lemma 1.3 (Amplitude of the first base). bσ 1E(1)(σ) = 1+O(b1/b2)σ. Proof. Immediate from the previous factorization. Proposition 1.4 (Peeling and iteration).Set S⟨1⟩(σ):=S(σ)−b−σ 1=∑j≥2b−σ j. Applying Lemma 1.2 to S⟨1⟩yields b2, and so on. Inductively, the whole Bis recovered. In D(s) = ∑k≥1f(k)E(k)(s), the level-1amplitude equals f(1). Finite–difference estimators (practical). For any fixed ∆>0, define bb1(σ,∆):=exp−1 ∆log S(σ+∆) S(σ)−−−→ σ→∞b1, with quantitative bound logbb1(σ,∆)−log b1≤∑j≥2(b1/bj)σ1−(b1/bj)∆ ∆1+∑j≥2(b1/bj)σ. Estimate f(1)via b f1(σ):=bb1(σ,∆)σE(1)(σ), subtract b f1bb−σ 1from f(1)E(1)inside D, and iterate to obtain bb2,bb3, . . . . If f(1) = 0, start at the smallest k≥2with f(k)=0, apply the same procedure to E(k), and divide recovered exponents by k. Cross–checks from higher fibers. For k≥2, E(k)(σ+∆) E(k)(σ)=b−k∆ 11+O(b1/b2)σ,bkσ 1E(k)(σ) = 1+O(b1/b2)σ. 4 Zeta via Value Fibers (Creating Irreducibles) Fix ℜs>1and set ζ(s) = ∞ ∑ n=1 n−s,D(s):=log ζ(s). Since D′(s) = ζ′(s) ζ(s)=−G(s),G(s) = ∑ n≥1 a(n)n−s, we have the Dirichlet–convolution identity G(s)ζ(s) = −ζ′(s).(1) Integrating termwise (legitimate by absolute convergence for ℜs>1) and using limσ→+∞D(σ) = limσ→+∞log ζ(σ) = 0(since ζ(σ)→1), gives D(s) = Z+∞ sG(σ)dσ=∑ n≥2 a(n) log nn−s. Define the value fibers Ey(s):=∑ n≥2 a(n)/ log n=y n−s,D(s) = ∑ y y Ey(s). If exp D(s)admits a uniform local factor F(z) = exp∑k≥1f(k)zkas in Theorem 1.1, then necessarily y=f(k)and Ef(k)(s) = ∑b∈Bb−ks for a tower primitive base set B. Moreover, because exp D(s) = ζ(s), comparing the coefficient of the pure powers b−ks (which can only arise by taking the k-th term from the single factor F(b−s)and constants from all others) forces [zk]F(z) = 1for all k≥0, hence F(z) = ∑ k≥0 zk=1 1−zand f(k) = 1 k. Therefore exp D(s) = exp ∑ k≥1 1 kE1/k(s)=∏ b∈B exp ∑ k≥1 1 kb−ks=∏ b∈B 1 1−b−s, i.e. ζ(s) = ∏b∈B(1−b−s)−1, and the bases B(constructed irreducibles) can be recovered by slope–peeling from the level fibers (Section: Recovering the Bases). Lemma 1.5 (Coefficient inversion).There is a unique sequence {a(n)}such that for all m≥1, ∑ d|m a(d) = log m.(2) Equivalently, a(1) = 0, a(m) = log m−∑ d|m 1≤d<m a(d) (m≥2).(3) Proof. Comparing coefficients in (1) yields ∑d|ma(d) = log m. Uniqueness follows from the triangular recursion (3). 5 Consequences for value fibers. With f(k) = 1/kdetermined above (by single–tower coefficient matching), the nonzero value fibers are exactly E1/k(s) = ∑ b∈B b−ks (k≥1), with no aliasing since the values 1/kare distinct. Thus Theorem 1.1 applies verbatim and yields the Euler product with local factor F(z) = (1−z)−1. Remark (Unbundling vs. identification).Coalgebra–primitivity selects the irreducible type, but the exponential map bundles each base into a local factor F(z) = (1−z)−1, so the observable data are the level fibers E(k)(s) = ∑b∈Bb−ks. The values 1/kensure no inter–level aliasing, yet bases are bundled within each level. Identifying the actual bases b∈Btherefore requires an unbundling step (slope–peeling) from E(1), after which the classical primes appear as the recovered primitive bases (tower primitive = coalgebra primitive) in N. Remark (Why ζdoes not directly factor integers).The Euler product shows that ζencodes multiplicative structure, but each base appears bundled as a whole tower via (1−b−s)−1. The values 1/kprevent inter-level aliasing, yet bases remain aggregated within each level fiber E(k). Identifying the actual bases requires an unbundling procedure (slope–peeling of E(1), or, analytically, coefficient extraction from −ζ′/ζ). Thus ζreconstructs the irreducibles globally but does not, by itself, provide a direct per-integer factorization operator. Once one introduces standard arithmetic functions (Möbius, von Mangoldt, divisor functions), each brings its own limitations for factorization, which can blur whether the bottleneck comes from ζ’s intrinsic bundling or from the auxiliary function used. Concretely—Euler product explains structure, not computation; it reconstructs irreducibles globally but is not a per–integer factorization oracle. Diagnostics for Failure of Tower–Invariant Decomposition If one attempts the above value–fiber decomposition but the value–fiber data do not satisfy the tower–invariance criterion (Theorem 1.1), the process breaks down in one of two ways: •Non–tower fibers. There exists a fiber level yfor which Sy={n≥2 : a(n)/ log n=y} does not consist entirely of perfect kth powers for any single k. Equivalently, no single exponent ksatisfies n=bkfor all n∈Sy, so one cannot form a consistent tower {bk:b∈ B}. •Inconsistent base sets. Even if each fiber Sf(k)happens to be composed of kth powers, the extracted base–sets Bk={n1/k:n∈Sf(k)} depend on krather than coincide with a single uniform set B. In either scenario, there is no factorization of the uniform local–factor form exp D(s) = ∏b∈BF(b−s): the value–fiber data fail tower–invariance, so the algorithm terminates without recovering a base set or a uniform local factor. Pipeline. Value fibers ⇒tower–invariance ⇒slope–peeling recovers B⇒F(z) = exp(∑k≥11 kzk) = (1−z)−1⇒ζ(s) = ∏b∈B(1−b−s)−1. (No unique factorization or primes are assumed; Bis constructed.) 6 Corollary 1.6 (Late identification of the objects).From the value–fiber decomposition of D(s) = log ζ(s)and the uniform local–factor (tower–invariance) criterion, one uniquely reconstructs the tower primitive base set Band the weights f(k) = 1/k(e.g. by slope–peeling the level–1fiber). Consequently, exp D(s) = ∏ b∈B F(b−s)with F(z) = 1 1−z, so ζ(s) = ∏ b∈B (1−b−s)−1. In Nthese tower primitive bases coincide with the usual multiplicative irreducibles (classical primes), but no such objects are assumed a priori: they are recovered from (2) together with the uniformity criterion. Remark (What was, and was not, used).We never invoked any pre–existing functions tied to prime factorization. The sequence a(n)was defined solely by the divisor–sum identity (2), i.e. by invertibility of the divisor–sum operator in the Dirichlet algebra. The uniformity condition then forces the tower–invariant structure, from which Band Fare reconstructed. Standard arithmetic functions (Möbius, von Mangoldt, divisor functions) can be introduced, but they are not required here: the multiplicative irreducibles (primes) are constructed from scratch as the outcome of the value–fiber data plus the uniform local–factor constraint. Coderivation and the von Mangoldt Function We did not use classical arithmetic functions to construct the bases, but they arise naturally inside the same Hopf framework. Coderivation. Define ∂:=−d ds ,∂n−s= (log n)n−s. Then ∂is a coderivation for the Dirichlet coalgebra ∆(n−s) = ∑d|nd−s⊗(n/d)−s, since ∆ ∂(n−s)=∑ d|n (log n)d−s⊗(n/d)−s =∑ d|nlog d+log(n/d)d−s⊗(n/d)−s = (∂⊗id +id ⊗∂)∆(n−s). Logarithmic coderivative of a group-like. Because ζ(s) = ∑n≥1n−sis group-like (∆ζ=ζ⊗ ζ), ∆∂ζ= (∂ζ)⊗ζ+ζ⊗(∂ζ), so the logarithmic coderivative L∂(ζ):=ζ−1(∂ζ) where ζis invertible in the completed algebra (since ζ(s) = 1+∑ n≥2 n−s) and coincides with the usual logarithmic derivative in s: L∂(ζ)(s) = −ζ′(s) ζ(s)=∑ n≥1 a(n)n−s(ℜs>1). and in multiplicative Hopf models (where ∆is an algebra map) it is coprimitive. Here, under the divisor–sum coalgebra, we do not assert coprimitivity. 7 The coefficients a(n)are uniquely determined by the divisor–sum relation ∑ d|m a(d) = log m(m≥1), i.e. by the same triangular inversion used earlier (no Möbius is required to state or solve this). In classical notation one writes a(n) = Λ(n). Per–prime view (optional). In the equivalent per–prime divided–power Hopf algebra (where each gp= (1−p−s)−1is group-like and log gpis primitive), −∂log gp=∑ k≥1 (log p)p−ks, and summing over preproduces −ζ′/ζ=∑na(n)n−s. Thus the Hopf structure both explains the Euler product (group-likes from primitives) and recovers the “von Mangoldt” coefficients via a canonical coderivation, with no extra ingredients. Examples and Non–Examples 1. Dirichlet from a coprime generator set (no primes assumed) Let M⊂ {2, 3, . . . }be pairwise coprime and assume ∑m∈Mm−σ<∞for some σ>1. Let ⟨M⟩:={∏imei i:ei∈Z≥0}be the multiplicative semigroup generated by M(which is divisor–closed because the miare pairwise coprime), and define a(n) = 1n∈⟨M⟩. Then, for ℜs>σ, ∑ n≥1 a(n) ns=∏ i∑ e≥0 m−es i=∏ i 1 1−m−s i . Here the bases are exactly the elements of M, recovered by slope–peeling from the level-1fiber E(1)(s) = ∑m∈Mm−s(tower primitive normalization implicit). 2. Concrete non–homogeneous example (no uniform local factor) Fix disjoint sets B1,B2⊂ {2, 3, . . . }of tower primitive bases (none a perfect power), and assume ∑b∈B1b−σ+∑c∈B2c−σ<∞for some σ>1. Define, for k≥1, a(n) =        1 k,n=bkwith b∈B1, 2 k,n=ckwith c∈B2, 0, otherwise. Then the value fibers split as E1/k(s) = ∑ b∈B1 b−ks,E2/k(s) = ∑ c∈B2 c−ks. Any factorization of the form exp ∑ n≥1 a(n)n−s=∏ b∈B F(b−s) would require a uniform local factor F(z) = exp ∑k≥1f(k)zkindependent of the base. But here the per–base “Bell series’’ differ: for b∈B1one needs [zk]log F(z) = 1/k(so Fb(z) = (1−z)−1), whereas for c∈B2one needs [zk]log F(z) = 2/k(so Fc(z) = (1−z)−2). Thus no single Fworks across all bases; the criterion of Theorem 1.1 fails. 8 3. ζ(s)2: two viewpoints (a) Log–based, passes uniformly. If we start from D(s) = 2 log ζ(s), then exp D(s) = ζ(s)2=∏ b∈B (1−b−s)−2, so the uniform local factor is F(z) = (1−z)−2with the same base set B. This is just the zeta case with weights doubled; tower–invariance holds. (b) Series–based, fails tower–invariance. If instead we take the Dirichlet series with divisor function D(s) = ∑ n≥1 d(n)n−s(ℜs>1), and ask whether exp D(s) = ∏b∈BF(b−s)with a uniform F(z) = exp(∑k≥1f(k)zk), the value fibers mix towers and composites: d(pk) = k+1, d(pq) = 4, d(p3) = 4. Thus the fiber S4={n:d(n) = 4}contains both p3(a pure k=3tower) and pq (product of two distinct bases), so it is not a single kth-power tower. Hence the tower–invariance criterion fails (non–tower fibers), and no uniform local factor exists for exp∑d(n)n−s. Without verifying each step of the pipeline, there is no a priori guarantee that ζitself does not fail at some stage. For reassurance, note that the logarithmic derivative passes all stages in the half–plane ℜs>1: −ζ′ ζ(s) = ∑ n≥1 Λ(n)n−s,Λ(n) log n=(1/k,n=pk, 0, otherwise. Hence the nonzero value fibers are exactly E1/k(s) = ∑ p p−ks (k≥1), with no inter–level aliasing; in particular E(1)(s) = ∑pp−s, so slope–peeling recovers the bases. Exercise 1.7 (Two hurdles: scaling and half–shift). (i) Apply the pipeline to ζ(2s). Show that the uniform local factor is F2(z) = exp(∑k≥1z2k/k) = (1−z2)−1and explain the identifiability ambiguity: peeling E(2) naively returns the squared bases {b2}. Resolve it by tower primitive normalization: detect the minimal nonzero level k0=2and take k0–th roots of the peeled exponents to recover the primitive bases. (ii) Consider the half–shift (Hurwitz) ζ(s,1 2) = ∑n≥0(n+1 2)−s=2s(1−2−s)ζ(s). Explain why the factor 2slies outside the Dirichlet algebra spanned by {n−s}, and why the uniform local–factor decomposition does not apply verbatim. Show that restricting to odd indices (i.e., (1−2−s)ζ(s)) restores the same base set but only over odd n. Algorithmic Reconstruction (finite data) Given the truncated data {(n,a(n)) : 2 ≤n≤N}and fixed parameters σ≫1,∆>0, proceed as follows. (If f(1)is not known a priori, identify the k=1fiber by selecting the value yfor which E(N) y(σ+∆)/E(N) y(σ)decays slowest in ∆; in the zeta case f(1) = 1.) 9