Deriving an Approximate Formula for Dirichlet L-Functions

Table of Links

1. Abstract & Introduction
2. Notation and outline of the proof
3. The set Ψ1
4. Zeros of L(s, ψ)L(s, χψ) in Ω
5. Some analytic lemmas
6. Approximate formula for L(s, ψ)
7. Mean value formula I
8. Evaluation of Ξ11
9. Evaluation of Ξ12
10. Proof of Proposition 2.4
11. Proof of Proposition 2.6
12. Evaluation of Ξ15
13. Approximation to Ξ14
14. Mean value formula II
15. Evaluation of Φ1
16. Evaluation of Φ2
17. Evaluation of Φ3
18. Proof of Proposition 2.5

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