The Riemann Hypothesis (educational)

For complex s with Re(s) > 1, the Riemann zeta function is defined by the convergent series:

ζ(s) = Σ_{n=1..∞} 1 / n^s

This function extends (by analytic continuation) to almost all complex numbers, with a simple pole at s = 1.

ζ(s) has “trivial” zeros at negative even integers. The “nontrivial” zeros lie in the critical strip:

0 < Re(s) < 1

Riemann Hypothesis (RH): Every nontrivial zero of ζ(s) has real part exactly 1/2.

ζ(s) = 0 and 0 < Re(s) < 1  ⇒  Re(s) = 1/2

Riemann discovered a deep link between the zeros of ζ(s) and the distribution of prime numbers. Informally: the closer the zeros are to the critical line, the more regular the error terms become when you approximate prime-counting functions. RH is considered one of the most important open questions in analytic number theory.

• Reproducible numeric experiments in Notebook (store assumptions + intermediate steps).
• Complex arithmetic (rect + polar, cis, conj, arg, re/im) for exploring ζ-like series.
• Explainable outputs: normalization + evaluation traces help catch subtle input mistakes.
• Offline-first: your lab notes survive flaky connectivity.

# ζ(2) should equal π²/6
zeta2 = zeta(2)
zeta2_40 = zeta(2; 40)
pi2_over6 = pi^2/6
zeta2_40 - pi2_over6

# η(1) = ln(2)
eta1 = eta(1)
eta1 - ln(2)

z = 3 + 4*i
abs(z)
arg(z)
conj(z)

# A point on the critical line Re(s)=1/2
s = 0.5 + 14.134725*i
zeta(s)       # approximate (truncated)
abs(zeta(s))

This is an interactive visualization of log₁₀|f(s)| over a rectangle in the complex plane, where s = x + i y. Dark/bright regions indicate where the magnitude is small/large (auto-scaled). The critical line Re(s)=1/2 is drawn as a vertical guide.

Implementation note: this plot uses the same truncated Hasse-series backend as the built-in eta(s) and zeta(s) functions. It’s an intuition tool — not a proof engine.