Problem 2 - Complex Analysis

Question

The FOURIER coefficients of the normalized EISENSTEIN series $$ \frac{(k-1) !}{2(2 \pi i)^{k}} G_{k}(z)=\sum_{n=0}^{\infty} a(n) e^{2 \pi i n z}, \quad k \geq 4 $$ satisfy the equations (a) $$ \begin{aligned} a(n) a(m) &=a(n m), & \text { if }(n, m)=1 \\ a\left(p^{\nu+1}\right) &=a(p) a\left(p^{\nu}\right)-p^{k-1} a\left(p^{\nu-1}\right) \end{aligned} $$ Deduce from this $$ \begin{aligned} \sum_{n=1}^{\infty} a(n) n^{-s} &=\prod_{p} \sum_{\nu=0}^{\infty} a\left(p^{\nu}\right) p^{-\nu s} \\ &=\prod_{p} \frac{1}{\left(1-p^{-s}\right)\left(1-p^{k-1-s}\right)} \\ &=\zeta(s) \zeta(s+1-k) \text { for } \sigma>k \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( \zeta(s) \zeta(s+1-k) \) for \( \sigma > k \).

Step 1: Understand the Fourier Series

The Fourier series given is a normalized Eisenstein series of weight \( k \). The series is represented as \( \frac{(k-1)!}{2(2 \pi i)^{k}} G_{k}(z) = \sum_{n=0}^{\infty} a(n) e^{2 \pi i n z} \). The coefficients \( a(n) \) generate functions in the series, given by conditions (a).

Step 2: Analyze the Conditions for Coefficients

We have two conditions: \( a(n) a(m) = a(nm) \) if \( (n, m) = 1 \), which indicates multiplicative behavior, and \( a(p^{u+1}) = a(p)a(p^{u}) - p^{k-1} a(p^{u-1}) \), a recursive relation related to prime powers.

Step 3: Setup the Dirichlet Series

Denote the Dirichlet series \( \sum_{n=1}^{\infty} a(n) n^{-s} \). Using multiplicativity, express \( a(n) \) terms as products of their prime power contributions.

Step 4: Apply the Euler Product Formula

Using the multiplicative property, express \( \sum_{n=1}^{\infty} a(n) n^{-s} \) as a product over all primes: \( \prod_{p} \sum_{u=0}^{\infty} a\left(p^{u}\right) p^{-u s} \). The recursive relation simplifies further as we calculate for each prime.

Step 5: Simplify the Product

From the recursive relation \( a(p^{u+1}) = a(p) a(p^{u}) - p^{k-1} a(p^{u-1}) \), derive \( \sum_{u=0}^{\infty} a\left(p^{u}\right) p^{-u s} = \frac{1}{\left(1-p^{-s}\right)\left(1-p^{k-1-s}\right)} \).

Step 6: Conclusion With Riemann Zeta Functions

The simplified product leads to the identity \( \prod_{p} \frac{1}{\left(1-p^{-s}\right)\left(1-p^{k-1-s}\right)} = \zeta(s) \zeta(s+1-k) \), valid for \( \sigma > k \).

Key Concepts

Fourier coefficients

The concept of Fourier coefficients arises in the context of Fourier series, which are used to represent periodic functions by decomposing them into sums of simpler sine and cosine terms. In the case of Eisenstein series, which are a special type of function important in number theory and modular forms, the Fourier coefficients take on a key role. These coefficients, denoted as \( a(n) \), serve as the building blocks for the series representation of the Eisenstein series.
The normalized Eisenstein series can be written as \( \frac{(k-1)!}{2(2 \pi i)^{k}} G_{k}(z) = \sum_{n=0}^{\infty} a(n) e^{2 \pi i n z} \). Each coefficient \( a(n) \) has properties that allow us to analyze the series deeply. For instance:

• If \( n \) and \( m \) are coprime, the coefficients are multiplicative: \( a(n) a(m) = a(nm) \).
• The recursive relation \( a(p^{u+1}) = a(p) a(p^{u}) - p^{k-1} a(p^{u-1}) \) describes how coefficients are related for powers of a prime \( p \).

These properties aid in further analysis, opening pathways to expressing the series in terms of Dirichlet series and linking it to the Riemann Zeta function.

Dirichlet series

Dirichlet series are functions that arise naturally when extending our understanding of the Fourier coefficients of particular functions, such as Eisenstein series. A Dirichlet series is typically expressed as \( \sum_{n=1}^{\infty} a(n) n^{-s} \), where \( a(n) \) are coefficients that carry significant information about the function. In this case, they relate to the behavior of the Eisenstein series.
The concept of the Dirichlet series becomes particularly useful with its property of convergence in a half-plane of complex numbers, facilitating deep connections between number theory and analytic functions. In the exercise, by employing the properties of the Fourier coefficients, we construct the Dirichlet series for Eisenstein series:

• The multiplicativity conditions help in expressing the Dirichlet series as an Euler product: \( \prod_{p} \sum_{u=0}^{\infty} a\left(p^{u}\right) p^{-u s} \).
• The recursive relation helps break down each term in the product, simplifying the infinite sum over powers \( u \).

This powerful representation aids in revealing deeper connections to other mathematical functions, such as the Riemann Zeta function.

Riemann Zeta function

The Riemann Zeta function, \( \zeta(s) \), is a central object in number theory and complex analysis, defined as \( \zeta(s) = \sum_{n=1}^{\infty} n^{-s} \) for complex numbers \( s \) with a real part greater than 1. It extends the notion of the harmonic series and plays a crucial role in understanding distribution of prime numbers.
In the study of modular forms and Eisenstein series, the Riemann Zeta function emerges naturally. Through the formulation of the Dirichlet series and its expression as an Euler product, a beautiful identity involving zeta functions is obtained in our context:

• For the Eisenstein series, this relationship simplifies to \( \prod_{p} \frac{1}{\left(1-p^{-s}\right)\left(1-p^{k-1-s}\right)} = \zeta(s) \zeta(s+1-k) \).
• This identity holds for \( \sigma > k \), thereby linking the infinite product representation directly with zeta functions.

Such relationships highlight the interplay between modular forms, number theory, and complex analysis, showcasing the Riemann Zeta function's role as a connecting thread across various mathematical disciplines.