Problem 2 - Complex Analysis

Question

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

Step 1: Understanding the Eisenstein Series

For the normed Eisenstein series given by \( \frac{(k-1)!}{2(2\pi i)^k} G_k(z) = \sum_{n=0}^{\infty} a(n) e^{2 \pi \ln x} \), \(G_k(z)\) is a holomorphic function that is related to the Riemann Zeta function by its Fourier expansion. The coefficients in the series must satisfy certain properties which are often related to number theory, specifically involving multiplicative functions.

Step 2: Analyzing the Properties of Coefficients

The coefficients \(a(n)\) satisfy two key recursion relations: for relatively prime numbers \(n\) and \(m\), \(a(n) a(m) = a(nm)\), and for powers of primes \(p^{u}\), \(a(p^{u+1}) = a(p)a(p^{u}) - p^{k-1}a(p^{u-1})\). These relations suggest that \(a(n)\) is a multiplicative function, particularly relating to divisors of integers.

Step 3: Expressing the Dirichlet Series

The goal is to sum up these coefficients in the form of a Dirichlet series: \( \sum_{n=1}^{\infty} a(n) n^{-s} \). Given the multiplicative nature, this becomes a product over primes: \( \prod_{p} \sum_{u=0}^{\infty} a(p^{u}) p^{-u s} \). By examining how \(a(p^u)\) behaves, specifically that it's governed by equations we have, each component can be translated using its recursive relation.

Step 4: Simplifying the Infinite Product

For each prime \(p\), transform the series \( \sum_{u=0}^{\infty} a(p^u) p^{-u s} \) into a format that encodes both the multiplicative nature and the influence of the recursive formula. Notice that this forms a geometric series resembling the Euler product in the zeta function.

Step 5: Deriving the Zeta Function Product

Recognizing that the product takes the form of products of factors \(\frac{1}{(1-p^{-s})(1-p^{k-1-s})}\), identify these as the factors corresponding to the Riemann Zeta function \( \zeta(s) \) and \( \zeta(s+1-k) \). These arise due to the similarities with Euler's product formula for zeta functions.

Key Concepts

Fourier coefficients

Fourier coefficients are crucial components in the study of periodic functions. They emerge in the Fourier series, a way to represent a function as a sum of sine and cosine terms. When dealing with Eisenstein series, which are specialized series in mathematics, the Fourier coefficients still retain their importance. Here, they help to express the series in terms of exponential functions, allowing for a deeper analysis of the function's properties.

• A Fourier coefficient of a function gives us information on how much a specific frequency component is present in the original function.
• In the context of Eisenstein series, these coefficients are not determined merely by basic sine and cosine waves but are related to more complex mathematical constructs.

Understanding the properties of these coefficients can reveal intricate relationships in number theory, including how certain types of numbers, like primes, interact through these series.

Dirichlet series

Dirichlet series are important mathematical expressions used in number theory. They take the form \( \sum_{n=1}^{\infty} a(n) n^{-s} \), which represents an infinite series where each term involves the coefficients \(a(n)\) and a variable \(s\). These series are incredibly useful because they allow functions to be explored in terms of convergence and analytic properties.

• Each Dirichlet series can be linked to a function unique to its constituents, such as how the coefficients \(a(n)\) interact.
• By examining these coefficients within the series, mathematicians extrapolate information about the distribution of numbers, like primes, and their properties.

Dirichlet series often appear in the study of L-functions and modular forms, broadening their utility in advanced mathematical research.

Multiplicative functions

Multiplicative functions are types of functions important in number theory. These functions satisfy the property that the function evaluated at the product of two coprime numbers is equal to the product of the function evaluated at each number individually.

• The Eisenstein series coefficients are examples, as shown in the condition that \(a(n)a(m) = a(nm)\), where \((n, m) = 1\).
• This property makes multiplicative functions especially useful when dealing with products and divisors, underpinning various number-theoretic results.

Multiplicative functions enable researchers to simplify complex arithmetic functions, helping to find patterns and predict behaviors in mathematical series and sequences.

Riemann Zeta function

The Riemann Zeta function, \(\zeta(s)\), is a cornerstone of number theory. It is crucial in understanding the distribution of prime numbers through its connection to Euler's product form for the zeta function.

• In the context of Eisenstein series, knowing the relation \( \zeta(s) \zeta(s+1-k) \) involves connecting complex analysis with number theory.
• This function illustrates how infinite series and products converge or diverge, providing insights into the convergence regions and residue calculations.

The Riemann Zeta function has profound implications, most famously related to the unsolved Riemann Hypothesis, which proposes that all non-trivial zeros lie on a specific line in the complex plane.