How Would I Approach Evaluating The Poisson Summation Of A Theta Function With A Dirichlet Character Convolution Twist?

Feb 28, 2025 by ADMIN 120 views

Introduction

In the realm of analytic number theory, the Poisson summation formula is a powerful tool for evaluating the Fourier transform of a function. When combined with the convolution of Dirichlet characters, it can lead to fascinating results. In this article, we will delve into the evaluation of the Poisson summation of a theta function with a Dirichlet character convolution twist.

Background and Notation

Before we dive into the evaluation, let's establish some notation and background.

Dirichlet Characters: A Dirichlet character is a multiplicative function χ: ℤ → ℂ that satisfies χ(1) = 1 and χ(mn) = χ(m)χ(n) for all m, n ∈ ℤ. Dirichlet characters are used to study the distribution of prime numbers and other arithmetic properties of integers.
Convolution: The convolution of two functions f and g is defined as (f ∗ g)(n) = ∑_{k ∈ ℤ} f(k)g(n - k).
Theta Function: A theta function is a function of the form θ(z) = ∑_{n ∈ ℤ} e^{2πinz}, where z is a complex number.

The Poisson Summation Formula

The Poisson summation formula states that for a function f ∈ L^1(ℝ), we have:

∑{n ∈ ℤ} f(n) = ∑{n ∈ ℤ} \hat{f}(n), where \hat{f}(n) = ∫_{-∞}^∞ f(x)e^{-2πinx}dx.

Evaluating the Poisson Summation of the Given Series

We are given the series:

∑_{n = -∞}^∞ (χ_1 ∗ χ_2)(n)e^{-πn2z}, where χ_1 and χ_2 are Dirichlet characters.

To evaluate this series, we can use the Poisson summation formula. However, we need to find the Fourier transform of the function (χ_1 ∗ χ_2)(n)e^{-πn2z}.

Finding the Fourier Transform

To find the Fourier transform of the function (χ_1 ∗ χ_2)(n)e^{-πn2z}, we can use the following formula:

\hat{f}(n) = ∫_{-∞}^∞ f(x)e^{-2πinx}dx.

In our case, f(x) = (χ_1 ∗ χ_2)(x)e^{-πx2z}.

Using the Convolution Property

The convolution property states that the Fourier transform of a convolution is equal to the product of the Fourier transforms.

\hat{(f ∗ g)}(n) = \hat{f}(n)\hat{g}(n).

In our case, we have:

\hat{(χ_1 ∗ χ_2)}(n) = \hat{χ_1}(n)\hat{χ_2}(n).

Finding the Fourier Transforms of the Dirichlet Characters

To find the Fourier transforms of the Dirichlet characters, we can use the following formulas:

\hat{χ_1}(n) = ∑{a ∈ ℤ} χ_1(a)e^{-2πina} and \hat{χ_2}(n) = ∑{b ∈ ℤ} χ_2(b)e^{-2πinb}.

Using the Poisson Summation Formula

Now that we have found the Fourier transforms of the Dirichlet characters, we can use the Poisson summation formula to evaluate the given series.

∑{n = -∞}^∞ (χ_1 ∗ χ_2)(n)e^{-πn2z} = ∑{n ∈ ℤ} \hat{(χ_1 ∗ χ_2)}(n)e^{-πn2z}.

Simplifying the Expression

Using the convolution property and the formulas for the Fourier transforms of the Dirichlet characters, we can simplify the expression:

∑{n ∈ ℤ} \hat{(χ_1 ∗ χ_2)}(n)e^{-πn2z} = ∑{n ∈ ℤ} \hat{χ_1}(n)\hat{χ_2}(n)e^{-πn2z}.