Prime counting functions, such as the Chebyshev function ψ(x; q, a), are inherently discontinuous step functions, jumping at prime powers pk ≡ a (mod q). The explicit formula connects these functions to the zeros of Dirichlet L-functions, providing a "spectral" representation analogous to how a Fourier series represents a periodic function using sines and cosines.
However, approximating discontinuous functions with sums of smooth functions (like xρ/ρ from the explicit formula or sine waves in Fourier series) inevitably introduces artifacts near the discontinuities. One well-known example is the Gibbs phenomenon, where persistent overshoots and undershoots occur near jumps, even when many terms are included. While such local artifacts are expected in the explicit formula approximation, the MAS framework focuses on a different, potentially much larger-scale distortion.
The concept of "Modular Aliasing" arises from this approximation context. Inspired by signal processing, where aliasing means misrepresentation of frequencies due to sampling issues or interference, modular aliasing describes how a hypothetical L-function zero ρ0 off the critical line (σ0 > 1/2) introduces a large, dominant error term (∝ xσ0). This term acts like a powerful, spurious "signal" that can systematically distort the perceived distribution of primes among different residue classes a (mod q), masking or mimicking true arithmetic patterns.
The Modular Aliasing Signature (MAS) hypothesis formalizes the specific characteristics of this distortion. While the framework offers valuable insights and testable predictions, it's crucial to understand what can be rigorously proven *within the framework's assumptions* versus what remains beyond its current scope, particularly concerning the proof of GRH itself.
The following results can be proven mathematically, solidifying the properties of the MAS model. These are conditional theorems, meaning they hold *under the explicit assumption* that GRH fails in the manner described (specifically, that a zero pair (ρ0, &bar;ρ0) with σ0 > 1/2 dominates the error term for large x).
Theorem (Form and Growth of MAS Error Term): Assume GRH fails for L(s, χ4) via a dominant zero pair (ρ0, &bar;ρ0) with σ0 > 1/2. Let ψE(x; 4, a) be the contribution of this pair to the error term in the explicit formula for ψ(x; 4, a). Then the amplitude envelope of ψE(x; 4, a) grows asymptotically as (1/|ρ0|) xσ0.
Proof Method: Direct analysis of the explicit formula term -(χ4(a)/2) * (xρ0/ρ0 + xρ0/ρ0) and the magnitude of complex exponentials (|xs| = xRe(s)).
Theorem (Anti-Correlation Property of MAS Error Term for q=4): Under the same assumptions, the error components for a=1 and a=3 satisfy ψE(x; 4, 1) = -ψE(x; 4, 3).
Proof Method: Direct algebraic consequence of substituting the character values χ4(1)=1 and χ4(3)=-1 into the definition of ψE(x; 4, a).
Theorem (Asymptotic Dominance of MAS Error Term):
Under the same assumptions, let ΔGRH(x; 4, a) be the error term ψ(x; 4, a) - x/φ(4) assuming GRH holds (known to be O(√x logO(1) x)). Then, for sufficiently large x, the MAS error term dominates: |ψE(x; 4, a)| / |ΔGRH(x; 4, a)| → ∞ as x → ∞.
Proof Method: Standard asymptotic analysis comparing the growth rate of xσ0 (with σ0 > 1/2) to x1/2 logk x for any fixed k.
These theorems provide a rigorous mathematical foundation for the specific signature predicted by the MAS hypothesis and observed in simulations. They represent new conditional knowledge about the precise nature of GRH failure.
It is equally important to recognize what the MAS framework, in its current form, cannot achieve:
Proof of GRH Itself: The MAS framework explores the consequences of GRH failure; it does not contain the necessary complex analytic machinery (related to the functional equation, properties of the L-function in the critical strip, zero-free region theorems, etc.) required to prove that zeros cannot exist off the critical line. It describes the symptoms of failure but cannot rule out the disease.
Derivation of a Definitive Mathematical Contradiction: While the MAS predicts significant distortions (like large oscillations potentially overwhelming Chebyshev's bias), proving that this distortion leads to an unavoidable contradiction with established mathematical truths is a major challenge. For instance:
While not directly provable from the MAS framework alone, its concepts might resonate with ideas from other advanced areas of mathematics and physics, suggesting deeper principles at play. These connections are currently speculative but offer avenues for future thought:
~√x). The MAS, driven by the large xσ0 term, suggests a less efficient encoding or the presence of a strong, structured, but "unwanted" signal component. Could GRH relate to a principle of minimal complexity or maximal compression for prime information, which is violated if a zero lies off-line?
Exploring these potential connections might offer deeper insights into why the critical line &Re(s)=1/2 appears so fundamental and why deviations (leading to MAS) seem problematic, even if formal proofs remain elusive.
Despite these limitations, the MAS framework offers significant value:
The MAS framework helps clarify *what GRH failure would look like*, potentially guiding future research—both computational searches and theoretical investigations within analytic number theory aimed at finding the elusive contradiction that would constitute a proof of GRH.
Author: 7B7545EB2B5B22A28204066BD292A0365D4989260318CDF4A7A0407C272E9AFB
The Modular Aliasing Signature
Numerical Test of the Aliasing Signature Framework for an Elliptic Curve L-function