Spectral Geometry and the One-Loop QED 𝛽-Function on 𝑆³×𝑆¹

1 Introduction

1. 1.1 Physical motivation and interpretation
2. 1.2 Connection to the spectral action principle
3. 1.3 Conventions and sign choices

2 Geometric Setup

1. 2.1 Metrics and curvature
2. 2.2 Gauge connection and Hopf bundle
3. 2.3 Twisted Dirac operator

3 Heat Kernel Expansion and the $a_{4}$ Coefficient

1. 3.1 General structure
2. 3.2 Bundle curvature decomposition
3. 3.3 Isolating the $F^{2}$ contribution

4 Mapping to the $\beta$-Function via $\zeta$-Regularization

1. 4.1 Effective action from the spectral zeta function
2. 4.2 One-loop correction to the gauge coupling
3. 4.3 The $\beta$-function

5 Discussion and Physical Interpretation

1. 5.1 Universality and parameter independence
2. 5.2 Limitations and the UV scale problem
3. 5.3 Comparison with related work

Geometric approaches to proving the Riemann hypothesis

Introduction: The Riemann Hypothesis and its Significance

The Riemann Hypothesis, first posited by Bernhard Riemann in his seminal 1859 paper, stands as one of the most profound and challenging unsolved problems in mathematics.[1, 2] At its core, the hypothesis makes a precise claim about the location of the nontrivial zeros of the Riemann zeta function, asserting that they all lie on the critical line in the complex plane where the real part of the complex variable s is equal to one-half.[3, 4, 5, 6] This seemingly simple statement has far-reaching consequences, particularly for our understanding of the distribution of prime numbers, those fundamental building blocks of arithmetic that cannot be expressed as the product of two smaller integers.[1, 3, 4, 5, 6, 7, 8, 9, 10] The hypothesis provides a crucial refinement to the Prime Number Theorem, which gives an asymptotic estimate of the density of primes. If the Riemann Hypothesis is true, it would establish a precise bound on the error term in this estimate, granting us a much deeper insight into the seemingly irregular pattern of prime numbers.[8, 11]