The Infinity Atlas: Axes, Charts and Finite Parts — from 0 + 0 + 0 + ⋯ to the Modular Surface

What does 1 + 2 + 3 + ⋯ really “equal”? Why does Euler’s constant appear where nothing seems to call for it? And why should the product of all prime numbers be 4π²?

The Infinity Atlas answers with one simple idea: infinity is not a number but an axis, and every way of approaching it is a chart. A divergent quantity becomes a point with coordinates on many levels: Ω², Ω, log Ω, and even complex levels N^ρ, one for each zero of the Riemann zeta function. Its finite part is one of those coordinates, and it has meaning once the chart is fixed.

The book is an illustrated field guide of 106 pages, with 45 plates and more than 25 verification tables. Its design mixes the look of an eighteenth-century treatise with holographic instrument panels. It climbs through five marked levels of difficulty, step by step, from school algebra to the modular surface.

What you will find

• A finite-part formula for any smoothed sum, and a clear rule for when finite parts are universal and when they depend on how the sum is cut off.
• The constants γ, ζ(2), ζ(3), … read as the price of changing chart. As a result ζ(3) appears in a law about primes alone. Explicit 1/log N corrections to generalized Mertens sums are verified up to 10¹⁰.
• The product of all primes, 4π², obtained as a residue rather than an axiom. It comes with new canonical values such as Σ_p (log p)² = −7.619049607…, and with Stokes constants given by odd zeta values (the first is 4π³/ζ(3)).
• The zeros of ζ seen as axes of prime sums, with measured Stokes jumps.
• A spectral chart λ = s(1 − s) for the Zeta Atlas. In it, the Laurent landscapes of the Eisenstein series converge to the centre E(z, ½). A gallery of 3D landscapes shows this.
• A status label on every statement (classical, proved, numerical, open), master tables of constants, and a MATLAB script that reproduces the main computations.

Who it is for

• Students who have met −1/12 and want to know what really lies behind it.
• Researchers in analytic number theory, asymptotics, resurgence and automorphic forms who want a compact map and statements they can test.
• Teachers looking for worked examples.
• Anyone curious about the infinite.

Foundations
Dirichlet series and the Mellin transform, Hankel’s formula, the Stokes phenomenon, the explicit formula of prime number theory, Montgomery’s theorem on minimal theta functions, and the companion volume The Zeta Atlas. Classical results are credited. The extremal law at Heegner points comes from the companion paper Euler’s constant in the Zeta Atlas.

An invitation
Chapter 12 sets out nine open problems at the edge of the map, each with what a solution would bring. They range from summing all the zero axes of the prime sum to a certified census of critical points and atlases in dimensions 8 and 24. You are warmly invited to check, correct, extend and build on any part of this work. Every number can be reproduced from the methods described.

This atlas is offered freely, with no commercial interest, as a contribution to the shared work of mathematics. Thank you for reading it, for your time, and for whatever you may build from it.

— Jorge Vicente Romero, October 2026

Computations and drafting were carried out with the assistance of an AI system (Claude, Anthropic); every number in the book was produced by executed code and checked as described in Appendix A.