PRH | Essay | 7.12 • Gaussian Blur, Gamma Factors, and the Functional Equation of the Riemann Zeta Function

Gaussian Blur, Gamma Factors, and the Functional Equation of the Riemann Zeta Function Aleksandar Perišić November 2025 Abstract The classical functional equation for the Riemann zeta function ξ(s) = ξ(1 −s) is usually presented as the outcome of a chain of analytic tricks: Poisson summation, theta functions, Mellin transforms and Gamma factors. In this note we reorganize that story from the perspective of blur. We view the Gaussian as a canonical blur kernel between the additive and multiplicative worlds. Its Mellin transform is the Gamma factor that appears in the completed zeta function; in this sense Γ( s/ 2) is the spectral remnant of the blur we have applied when probing the prime world. The functional equation becomes the statement that, after paying this blur cost, the resulting object is perfectly symmetric under s7→ 1−s. Finally, via the explicit formula, the symmetry line ℜ ( s ) = 1 / 2emerges as a natural uncertainty boundary: the critical line corresponds to square–root–scale fluctuations in prime counting, and no arrangement of zeros consistent with the functional equation can produce “less blur” than that. Zeros on or off the line only confirm that some irreducible blur is built into the system. 1 Introduction The Riemann zeta function ζ(s) = ∞ X n=1 1 ns,ℜ(s)>1, admits a meromorphic continuation to Cand satisfies the functional equation ξ(s) = ξ(1 −s), where the completed zeta is ξ(s) := 1 2s(s−1) π−s/2Γ s 2ζ(s). In this form, the Gamma factor Γ( s/ 2) and the power π−s/2 often look like technical decorations: necessary but mysterious. On the other hand, the blur method starts from a different attitude. One does not insist on perfectly sharp objects; instead one chooses a small, positive averaging kernel (a blur) and declares in advance how much unresolved structure one is willing to carry. Blur-invariant quantities are those that survive such averaging unchanged. The aim of this note is to explain how the classical derivation of the functional equation can be seen as a blur attack on zeta. The Gaussian kernel is precisely the blur we are inserting between addition and multiplication; the Gamma factor is the Mellin shadow of that blur; and the symmetry s↔ 1 −s expresses a kind of uncertainty tradeoff in the Mellin variable. In this view, the critical line ℜ ( s ) = 1 / 2is not just an axis of symmetry: it is the natural boundary of blur that cannot be escaped. 1 2 The classical functional equation in brief We recall the standard analytic mechanism just far enough to identify the Gaussian and the Gamma factor as a blur/probe pair. A more detailed treatment can be found, for instance, in [2,1]. 2.1 Theta, Gaussians, and Poisson summation Consider the Jacobi theta function θ(t) := X n∈Z e−πn2t, t > 0. Poisson summation and the self-duality of the Gaussian under Fourier transform give the key identity θ(t) = t−1/2θ(1/t). Here the Gaussian kernel e−πx2 plays two roles at once: it provides decay at infinity, and it is stable under the Fourier transform, which is what allows the clean t−1/2factor to appear. The theta function is a kind of generating function for the lattice Z , smeared by a Gaussian heat kernel. From the blur point of view, this is a first clear sign that we are probing the integers through a specific blur: a Gaussian on the additive side. 2.2 Mellin transform and the Gamma factor Next, we take a Mellin transform in tof θ(t)−1, which removes the constant term n= 0: I(s) := Z∞ 0θ(t)−1ts/2−1dt. By unfolding the definition of θ and interchanging sum and integral (justified in a right half-plane), we obtain I(s) = X n=0 Z∞ 0 e−πn2tts/2−1dt = 2 ∞ X n=1 Z∞ 0 e−πn2tts/2−1dt. The change of variables u=πn2tshows that each integral is Z∞ 0 e−πn2tts/2−1dt = (πn2)−s/2Γ s 2. Thus I(s) = 2 Γ s 2π−s/2 ∞ X n=1 1 ns= 2 π−s/2Γ s 2ζ(s), for ℜ ( s )large enough. In other words, the Mellin transform of the Gaussian-smeared theta function produces exactly the factor π−s/2Γ(s/2)ζ(s). At the same time, applying the theta identity θ ( t ) = t−1/2θ (1 /t )inside the integral and changing variables t7→ 1/t gives I(s)=I(1 −s), and comparing the two expressions yields the functional equation for ζ , in the completed form ξ(s)=ξ(1 −s). Everything here is standard. The only point we stress is: the Gaussian kernel in the theta function, and its Mellin transform Γ( s/ 2), are not cosmetic; they are the analytic bridge between the additive and multiplicative domains. 2 3 Gaussian blur and Gamma as a spectral remnant We now reinterpret the previous steps in blur language. The goal is not to change any formulas, but to clarify how closely the classical procedure matches a generic “blur attack” on an object. 3.1 Blur kernels on the line and on the log–line On the additive line R , a blur kernel is a positive, normalized function kτ ( x )depending on a scale parameter τ > 0, with ZR kτ(x)dx = 1, kτ→δ0 in the sense of distributions as τ↓ 0. Blurring a function f at scale τ means convolving with kτ : (Bτf)(x) = (kτ∗f)(x) = ZR kτ(x−u)f(u)du. Gaussian kernels kτ(x) = 1 √4πτ e−x2/(4τ) are the canonical example: they are positive, normalized, form a semigroup, and minimize the usual Fourier-analytic uncertainty relationship between position and frequency. On the multiplicative line (0 ,∞ )it is natural to blur in the log–variable. Changing variables x=euand using the Haar measure du/u leads to multiplicative blurs of the form (B× σf)(x) = Z∞ 0 Kσx uf(u)du u, where Kσis a blur kernel on the log–line. A log–Gaussian choice Kσx u=1 √2πσ exp−(log x−log u)2 2σ2 is a multiplicative analogue of the heat kernel. 3.2 Mellin transform as the spectral footprint of blur The Mellin transform of a function fis M[f](s) = Z∞ 0 f(x)xs−1dx. Applied to the blur kernel Kσ on the log–line, it produces a multiplier d Kσ ( s )in the s –plane. Under suitable assumptions one has M[B× σf](s) = d Kσ(s)M[f](s). Thus d Kσ ( s )is the spectral remnant of the blur: it encodes what the blur has done in the Mellin domain. In the classical zeta story the blur kernel is implemented at the level of the theta function as t7−→ e−πn2t, and the Mellin transform Z∞ 0 e−πn2tts/2−1dt = (πn2)−s/2Γ s 2 3 can be rewritten, via the change of variables t=u2, as Z∞ 0 e−πn2tts/2−1dt = 2 Z∞ 0 e−πn2u2us−1du, so it is literally the Mellin transform of the pure Gaussian u7→ e−πn2u2 . In particular, the spectral remnant of this Gaussian blur is exactly the Gamma factor Γ( s/ 2) together with the power π−s/2. That is: the completed zeta π−s/2Γ s 2ζ(s) is precisely what one obtains by probing the multiplicative structure with a Gaussian blur and then reading the result through the Mellin transform. In this sense, Γ( s/ 2) is not just “there because of the integrals”: it is the natural footprint of the Gaussian blur connecting the additive and multiplicative worlds. 4 The functional equation as symmetry under blur With this reinterpretation in place, the functional equation ξ(s)=ξ(1 −s) can be read as a statement about symmetry under a cannot-be-avoided blur between two complementary descriptions of the same object. 4.1 Additive vs. multiplicative descriptions On the additive side we have the theta function, built by blurring the integer lattice with a Gaussian in the t –variable. On the multiplicative side we have primes and the Dirichlet series for ζ ( s ). The theta identity θ ( t ) = t−1/2θ (1 /t )is a purely additive statement: it expresses the invariance of the blurred lattice under Fourier duality and scaling. After passing to Mellin, this additive identity becomes a relation between two multiplicative descriptions: π−s/2Γ s 2ζ(s)←→ π−(1−s)/2Γ 1−s 2ζ(1 −s). Multiplying by 1 2s ( s− 1) to clear the pole at s = 1 and the trivial zero at s = 0 gives the symmetric object ξ(s) = 1 2s(s−1) π−s/2Γ s 2ζ(s) with the clean symmetry ξ(s)=ξ(1 −s). In blur language: after we have paid the blur toll by inserting the Gaussian and accepting its spectral remnant Γ( s/ 2), the remaining object is perfectly symmetric between s and 1 −s . The blur has absorbed the asymmetry between the additive and multiplicative channels; what is left is an invariant that does not change when we swap the two. 4.2 The Mellin variable as frequency on the log–line There is a natural geometric way to see why s↔1−sresembles an uncertainty tradeoff. Writing x=eu, the Mellin kernel becomes xs−1dx =e(s−1)udu. Thus the Mellin variable s=σ+it plays the role of a frequency parameter on the log–line: 4 •σ=ℜ(s)controls exponential growth/decay in u; •t=ℑ(s)controls oscillation in u. The functional equation relates ξ ( s )and ξ (1 −s ), i.e. it ties together the values of ξ at frequencies σ + it and 1 −σ + it . The line ℜ ( s )=1 / 2is the fixed axis of this reflection: it is the “middle” between the two Mellin half-planes. Seen this way, ξ ( s ) = ξ (1 −s )says: once the Gaussian blur has been inserted, the spectral information in the half-plane ℜ ( s ) >1 2 is not independent of the spectral information in ℜ ( s ) <1 2 . Pushing precision in one half-plane automatically constrains what can happen in the other. The symmetry enforces a kind of balance between two complementary Mellin regimes. 5 The critical line as an uncertainty boundary The most concrete appearance of this balance is in the explicit formulas relating zeros of zeta to primes. These formulas make it clear why the line ℜ(s) = 1/2is a natural boundary of blur. 5.1 Zeros as sources of multiplicative noise In one standard form of the explicit formula, a prime-weighted counting function such as Chebyshev’s ψ(x)can be written as ψ(x) = x−X ρ xρ ρ+(other terms), where the sum runs over nontrivial zeros ρ = β + iγ of ζ ( s ), counted with multiplicity. The term xρ/ρ is an oscillatory contribution whose magnitude is governed by β=ℜ(ρ):  xρ ρ≈xβ |ρ|. Thus each zero behaves like a “noise source” on the multiplicative scale: the closer β is to 1, the larger its contribution to the fluctuations of ψ ( x ); the closer to 0, the smaller. The critical exponent β= 1/2corresponds to a square–root–scale contribution x1/2. 5.2 Symmetry of zeros and the square–root scale The functional equation ξ ( s ) = ξ (1 −s )implies that nontrivial zeros are symmetric with respect to the line ℜ ( s ) = 1 / 2: if ρ is a zero, then so is 1 −ρ . An off-line zero ρ = β + iγ with β > 1 / 2 comes with a partner 1−ρwith real part 1−β < 1/2. From the explicit formula one sees: • If all zeros lie on the critical line ℜ ( ρ ) = 1 / 2(the Riemann Hypothesis), then the oscillatory terms in ψ ( x )are naturally at the square-root scale: each contributes roughly x1/2 , and there are infinitely many such contributions. This provides an irreducible noise floor of order x1/2. • If some zeros lie off the line with ℜ ( ρ ) > 1 / 2, then their contributions grow like xℜ(ρ) , which is larger than x1/2 . The partner zero at 1 −ρ does not cancel this growth; symmetry ensures that whatever happens in one half-plane is mirrored in the other, but it does not remove the associated blur. In either case, the line ℜ(s) = 1/2marks a natural boundary: •If all zeros sit there, then square–root–scale blur is unavoidable. 5 •If any zeros move away from it, the blur only gets worse. There is no arrangement of zeros consistent with the functional equation that produces fluctuations substantially smaller than the square–root scale purely by spectral means. 5.3 Interpreting ℜ(s) = 1/2as minimal blur From the blur perspective, this suggests the following interpretation. • The Gaussian probe is the canonical blur kernel reconciling the additive and multiplicative descriptions; its Mellin transform Γ(s/2) is the spectral remnant we must carry. • The functional equation states that, after inserting this blur and passing to the completed zeta ξ(s), the spectral content is perfectly symmetric under s7→ 1−s. • The explicit formula then tells us how this symmetric spectral content manifests as multiplicative noise in prime counting. The line ℜ ( s )=1 / 2corresponds to square–root– scale fluctuations; no repositioning of zeros consistent with the symmetry can reduce the inherent blur below that scale. In this sense, we may say that ℜ ( s )=1 / 2is the natural blur boundary encoded in the zeta machinery. The blur can be as good as square–root scale, or worse, but not conceptually better while we respect the Gaussian/Gamma probe and the functional equation. 6 Conclusion Reconsidering the functional equation from the blur viewpoint brings several pieces of the classical picture under one roof: • The Gaussian kernel used in the theta function is a canonical blur: positive, normalized, and extremal for the Fourier uncertainty principle. • The Gamma factor Γ( s/ 2) and the power π−s/2 are exactly the Mellin transform of this blur. They are the spectral remnant of probing the primes through a Gaussian window on the additive side. • The completed zeta ξ ( s )is what remains after paying this blur cost. Its symmetry ξ ( s ) = ξ (1 −s )expresses the fact that, once the blur has done its job, the two complementary Mellin half-planes are tied together. • Via the explicit formula, the line ℜ ( s ) = 1 / 2emerges as an uncertainty boundary: it corresponds to square–root–scale fluctuations in prime counting, and no zero configuration compatible with the functional equation can avoid at least that much blur. Nothing in the analytic derivation changes. The contribution of the blur lens is conceptual: it explains why the Gaussian and Gamma appear exactly in the combination they do, and why the critical line is the natural place where the unavoidable blur of the zeta world lives. Thinking of π−s/2 Γ s 2 as the spectral signature of a Gaussian blur is not a radical reinvention, but rather a coherent naming of what the classical derivation is already doing. 6 References [1] H. M. Edwards, Riemann’s Zeta Function, Dover, 2001 (reprint of Academic Press, 1974). [2] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. HeathBrown, Oxford University Press, 1986. [3] A. Perišić, Blur as a Universal Principle: Number Theory, Probability, Dynamics, Zenodo, 2025. [4] A. Perišić, Soft-Addition and Soft-Multiplication and the Channel–Switch Error, Zenodo, 2025. [5] A. Perišić, Epistemological Blur, Zenodo, 2025. 7