Working definition
DLMF 25.2.1 defines the zeta function by the Dirichlet series, the sum of one over n to the s, and states it for real part of s greater than one. 25.2.11 gives the Euler product, the product over all primes of one minus p to the minus s, inverted, under the same condition. Elsewhere the function is defined by analytic continuation, and the chapter records that the continued function is meromorphic with a single singularity in the complex plane: a simple pole at s equals one with residue one. 25.2.4 gives the Laurent expansion about that pole, one over s minus one plus a series in the Stieltjes constants, which are themselves defined as limits at 25.2.5.
Notation
zeta of sthe product index p running over primesgamma sub n for the Stieltjes constantsAssumptions
- Both the series at 25.2.1 and the product at 25.2.11 are stated only for real part of s greater than one; neither defines the function elsewhere.
- Values outside that half-plane come from analytic continuation, which is a different operation from evaluating the series.
- The Laurent expansion at 25.2.4 is local to the pole at s equals one.
Invariants
- The Dirichlet series in the half-plane, DLMF 25.2.1
- The Euler product over primes in the same half-plane, DLMF 25.2.11
- A single simple pole at s equals one with residue one, DLMF 25.2 stated in prose rather than as a numbered equation
Reproducible procedure
- Check the real part of the argument before using either representation, since both stop at one.
- Treat a value at real part below one as a continuation rather than a sum, because the series diverges there.
- Near s equals one, use the Laurent form at 25.2.4 rather than the series, which is where the pole sits.
Error and boundary controls
- The Dirichlet series converges slowly near the boundary of its half-plane, so truncating it close to real part one gives poor accuracy long before it fails outright.
- The Euler product is over infinitely many primes; truncating it at a finite prime is an approximation the chapter does not bound.
- The Stieltjes constants at 25.2.5 are defined as limits of differences that cancel, so computing them naively loses significance.
What this does not establish
The chapter states representations and the pole. It settles no question about the location of the zeros, and neither representation defines the function outside its half-plane.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 25: Zeta and Related Functions · National Institute of Standards and Technology
Establishes: The Dirichlet series and Euler product for the Riemann zeta function with their shared half-plane of validity, the single pole of the continued function, and the Laurent expansion about it.
Boundary: The chapter states the representations and the analytic structure. The series and the product hold only where the real part exceeds one, and nothing here settles any open question about the zeros.
Direct answer
- DLMF 25.2.1 defines the zeta function by the Dirichlet series, the sum of one over n to the s, and states it for real part of s greater than one. 25.2.11 gives the Euler product, the product over all primes of one minus p to the minus s, inverted, under the same condition. Elsewhere the function is defined by analytic continuation, and the chapter records that the continued function is meromorphic with a single singularity in the complex plane: a simple pole at s equals one with residue one. 25.2.4 gives the Laurent expansion about that pole, one over s minus one plus a series in the Stieltjes constants, which are themselves defined as limits at 25.2.5.
Mechanism and method
- Check the real part of the argument before using either representation, since both stop at one.
- Treat a value at real part below one as a continuation rather than a sum, because the series diverges there.
- Near s equals one, use the Laurent form at 25.2.4 rather than the series, which is where the pole sits.
What is measured
- The Dirichlet series in the half-plane, DLMF 25.2.1
- The Euler product over primes in the same half-plane, DLMF 25.2.11
- A single simple pole at s equals one with residue one, DLMF 25.2 stated in prose rather than as a numbered equation
Limitations
- The Dirichlet series converges slowly near the boundary of its half-plane, so truncating it close to real part one gives poor accuracy long before it fails outright.
- The Euler product is over infinitely many primes; truncating it at a finite prime is an approximation the chapter does not bound.
- The Stieltjes constants at 25.2.5 are defined as limits of differences that cancel, so computing them naively loses significance.
- Both the series at 25.2.1 and the product at 25.2.11 are stated only for real part of s greater than one; neither defines the function elsewhere.
- Values outside that half-plane come from analytic continuation, which is a different operation from evaluating the series.
- The Laurent expansion at 25.2.4 is local to the pole at s equals one.
What this does not establish
- The chapter states representations and the pole. It settles no question about the location of the zeros, and neither representation defines the function outside its half-plane.