RiemannZetaZero

The $k$-th nontrivial zero of the Riemann zeta function on the critical line, $\tfrac12 + i t_k$ for a positive integer $k$. Wolfram calls this ZetaZero; compute-engine's own Fungrim identity table already spells it RiemannZetaZero. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimRiemannZetaZero↗Fungrim entry60c2ec↗Wolfram LanguageZetaZero!↗

RiemannZetaZero(k)the k-th nontrivial zero, $\tfrac12 + i t_k$.

Typed signatures

Type	Package	Applies
(integer) -> number	analytic	always

Domain: Special functions

Details

• $t_k$ is found by scanning RiemannSiegelZ for sign changes along the real line from $t \approx 0$, counting crossings as it goes, then bisecting the bracket once the $k$-th crossing turns up. This counts crossings directly rather than bracketing between Gram points, so it isn't exposed to a Gram's-law failure (the first is at Gram index 126) the way a Gram-point method would be.
• Supported range: $k$ such that $t_k \le 2000$ — comfortably past the first 1000 zeros, and well beyond every example here. Above that the call is left symbolic rather than guess further out with an unvalidated scan; $k < 1$ is likewise left symbolic, and a non-integer $k$ is a type error.
• Numeric only (N() or an inexact operand) — the real part is always exactly $\tfrac12$; the search cost is in the imaginary part.
• N(…, d) past a double's digits refines the double zero by Newton's method on the bignum RiemannSiegelZ.

Examples

1 more kept as data, checked against the oracles

Basic #

The first nontrivial zero, $\tfrac12 + 14.1347\ldots i$

live ↗

In$N(\mathrm{RiemannZetaZero}(1))$Out$0.5+14.134\,725\,141\,734\,695i$

live ↗

In$N(\mathrm{RiemannZetaZero}(2))$Out$0.5+21.022\,039\,638\,771\,556i$

Scope #

live ↗

In$N(\mathrm{RiemannZetaZero}(10))$Out$0.5+49.773\,832\,477\,672\,3i$

The first three zeros, all with real part $\frac12$

live ↗

In$N(\bigl\lbrack\mathrm{RiemannZetaZero}(1), \mathrm{RiemannZetaZero}(2), \mathrm{RiemannZetaZero}(3)\bigr\rbrack)$Out$\bigl\lbrack0.5+14.134\,725\,141\,734\,695i, 0.5+21.022\,039\,638\,771\,556i, 0.5+25.010\,857\,580\,145\,69i\bigr\rbrack$

live ↗

In$N(\mathrm{RiemannZetaZero}(54), 50)$Out$0.5+150.925\,257\,612\,241\,466\,761\,852\,524\,678\,305\,627\,602\,426\,770\,473i$

live ↗

In$N(\mathrm{RiemannZetaZero}(46), 20)$Out$0.5+134.756\,509\,753\,373\,871\,33i$

live ↗

In$N(\mathrm{RiemannZetaZero}(-1)=\overline{\mathrm{RiemannZetaZero}(1)})$Out$\mathrm{True}$

Possible issues #

Index 0 has no zero, so it stays unevaluated

live ↗

In$N(\mathrm{RiemannZetaZero}(0))$Out$\mathrm{RiemannZetaZero}(0)$

About

Examples	9
Checked against	Wolfram Language 8 agree1 differ
Crosswalk	FungrimFungrim entryWolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/riemann-siegel.ts

sign-change scan of RiemannSiegelZ + bisection; see the file header for the range this covers.

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

ZetaZero[k]; mpmath.zetazero(k).

See also: RiemannSiegelZ, Zeta