Deriving an Approximate Formula for Dirichlet L-Functionsby@eigenvalue

# Deriving an Approximate Formula for Dirichlet L-Functions

June 2nd, 2024

This section derives an approximate formula for L(s, ψ), leveraging the functional equation and contour integration, providing a detailed analysis with error estimation and key lemmas.

Author:

(1) Yitang Zhang.

Appendix A. Some Euler products

Appendix B. Some arithmetic sums

References

## 6. Approximate formula for L(s, ψ)

Write

Let

Lemma 6.1. Suppose ψ(mod p) ∈ Ψ, |σ − 1/2| < 2α and |t − 2πt0| < L1 + 2. Then

L(s, ψ) = K(s, ψ) + Z(s, ψ)N(1 − s, ψ¯) + O(E1(s, ψ)),

where

and where

Proof. By (4.3) we have

The left side above is, by moving the line of integration to u = −1, equal to

It therefore suffices to show that

For u = −1 we have, by the functional equation (2.2) with θ = ψ,

We first show that

We move the contour of integration in (6.2) to the vertical segments

and

with the horizontal connecting segments

whence (6.2) follows. The proof of (6.1) is therefore reduced to showing that

This paper is available on arxiv under CC 4.0 license.

L O A D I N G
. . . comments & more!