Understanding the Riemann Hypothesis

1. Is it axiomatic that all zeros lie on the critical line?

No, it is absolutely not axiomatic. It is the great open question.

The Riemann Hypothesis (RH) is a conjecture, not an axiom. An axiom is a statement that is taken to be true as the starting point for further reasoning.

The Riemann Hypothesis is the proposal that every non-trivial zero of the Riemann zeta function has a real part equal to 1/2. This statement is not assumed to be true. Mathematicians have been trying for over 160 years to:

• Prove it is true (elevating it to a theorem).
• Find a counterexample (a zero off the line within the critical strip), which would prove it false.

Many theorems are of the form "If the Riemann Hypothesis is true, then [a profound result] follows." This shows that RH is a powerful potential tool, not a foundational axiom.

2. Are zeros found by building interval to interval?

Yes, this is precisely how it's done computationally.

You've identified the primary method for verifying the hypothesis up to a certain height.

• The Argument Principle: A theorem from complex analysis that allows you to count the number of zeros of a function inside a given contour by calculating a specific integral.
• The Process:
  1. Define a rectangular region in the complex strip (e.g., with imaginary part from 0 to T).
  2. Use the argument principle to count the total number of zeros inside it.
  3. Computationally "chase" and locate each zero on the critical line with high precision.
  4. If the number of zeros found on the line matches the number proven to exist in the strip, then all zeros in that region lie on the critical line.

This is how we know the first 10¹³ (ten trillion) non-trivial zeros lie on the critical line. However, this is verification, not proof. A single counterexample in a higher interval would disprove the hypothesis.

3. How does expansion into infinity hinder a compact formula?

This is the core challenge of analysis. Infinity "hinders" a solution in several ways:

1. The Nature of Infinity: You can never check an infinite number of cases. A pattern can hold for the first trillion examples and then break. A counterexample could be lurking in the unimaginably large infinity.
2. Lack of a "Closed-Form" Solution: The Riemann zeta function is an infinite series. There is no known finite algebraic formula that neatly expresses its zeros.
3. The Need for Analytic Proof: We must prove the structure of the function forces all zeros onto the line. This requires proving deep, global properties like:
   • Symmetry: Zeros are symmetric around the critical line and the real axis.
   • Functional Equation: The function relates its value at s to its value at 1-s.

In summary: The "expansion into infinity" forces mathematicians to move from direct calculation to proving overarching structural theorems. The goal is not to find every zero but to prove that the function's very essence makes it impossible for any zero to deviate from the critical line. This is what makes the problem both immensely difficult and profoundly beautiful.