ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The character chi_4 and the Gregory-Leibniz series

Example

For the nonprincipal character $χ_{4}$ modulo $4$ ,

$L (1, χ_{4}) = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{π}{4} .$

Facts & Assumptions

Given: The definition of $L (s, χ)$ , the table for $χ_{4}$ , the holomorphic continuation of nonprincipal Dirichlet $L$ -functions, and the Gregory-Leibniz theorem (Dirichlet L-functions, Dirichlet character tables modulo 3, 4, and 5, Nonprincipal Dirichlet L-functions are holomorphic on Re s greater than 0, The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+...).

Verification

technique · direct

1.1

The unit group modulo $4$ is ${1, 3}$ , with $3^{2} \equiv 1 (m o d 4)$ . Its unique nontrivial character sends $1$ to $1$ and $3$ to $- 1$ , and extension by zero sends the even classes to $0$ . This is the character $χ_{4}$ listed in Dirichlet character tables modulo 3, 4, and 5, so $χ_{4} (2 m) = 0$ , $χ_{4} (4 m + 1) = 1$ , and $χ_{4} (4 m + 3) = - 1$ . The convergence theorem then identifies $L (1, χ_{4}) = \sum_{n \geq 1} \frac{χ_{4} (n)}{n} = \sum_{m \geq 0} (\frac{1}{4 m + 1} - \frac{1}{4 m + 3}) = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots .$

givenalgebra

2.1

The last series is exactly the Gregory-Leibniz series, so The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+... gives $L (1, χ_{4}) = π / 4$ .

step 1.1givenalgebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Kiran S. Kedlaya, Notes on Analytic Number Theory, section 3.2 (standard reference, not scraped)