8.1 Proof of non-vanishing of zeta function on Re(s) = 1
2 min read•august 9, 2024
The is a cornerstone of analytic number theory. Its on the line Re(s) = 1 is crucial for understanding prime number distribution and proving the .
This section dives into the proof of this non-vanishing property. We'll use contradiction, logarithmic derivatives, and to show why the zeta function can't be zero when the real part of s equals 1.
Riemann Zeta Function Properties
Definition and Convergence
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- Riemann zeta function defined as for complex s with Re(s) > 1
- Converges absolutely for Re(s) > 1 due to comparison with p-series
- Extends analytically to entire except for simple pole at s = 1
- relates values of zeta(s) to zeta(1-s)
Euler Product Representation
- expresses zeta function as product over primes: for Re(s) > 1
- Demonstrates deep connection between zeta function and prime numbers
- Provides key insight into distribution of primes
- Allows for alternative proofs of certain zeta function properties
Analytic Continuation and Critical Strip
- extends zeta function beyond its initial domain of convergence
- Achieved through various methods (Riemann's integral formula, Hankel contour)
- refers to vertical strip in complex plane where 0 ≤ Re(s) ≤ 1
- Contains all non- of zeta function
- conjectures all non-trivial zeros lie on Re(s) = 1/2
Non-vanishing on Re(s) = 1
Importance of Re(s) = 1 Line
- Real part of s equal to 1 forms boundary of absolute convergence for zeta function
- Non-vanishing on this line crucial for Prime Number Theorem
- Connects to behavior of primes through Euler product representation
- Studying zeta function on this line provides insights into distribution of primes
Proof Strategy and Key Concepts
- Non-vanishing proven by contradiction, assuming zeta(s) = 0 for some s with Re(s) = 1
- of zeta function plays central role in proof
- Defined as where ζ'(s) is derivative of zeta function
- Relates to sum of prime powers through Euler product formula
Dirichlet Series and Convergence
- Dirichlet series form generalizes zeta function
- Convergence of Dirichlet series depends on behavior of coefficients a_n
- For zeta function, all a_n = 1, leading to convergence for Re(s) > 1
- Proof uses properties of Dirichlet series to derive contradiction from assumed zero
Key Terms to Review (15)
Analytic Continuation: Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its original area of definition, allowing it to be expressed in a broader context. This process is crucial for understanding functions like the Riemann zeta function and Dirichlet L-functions, as it reveals their behavior and properties in different regions of the complex plane.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work laid foundational concepts in number theory, particularly with his introduction of the Riemann zeta function. His exploration of this function opened up pathways to understand the distribution of prime numbers and provided a critical link between analysis and number theory, shaping many essential properties and conjectures in modern mathematics.
Complex Plane: The complex plane is a two-dimensional geometric representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This visualization aids in understanding complex functions and their behavior, especially in the context of important mathematical concepts like convergence, continuity, and analytic properties.
Critical Line: The critical line refers to the vertical line in the complex plane defined by the equation Re(s) = 1/2, where s is a complex number. This line is significant in the study of the Riemann zeta function and its properties, particularly concerning the distribution of prime numbers and the famous Riemann Hypothesis.
Critical Strip: The critical strip refers to the vertical region in the complex plane where the real part of a complex variable s is between 0 and 1, specifically defined as the region where $$0 < Re(s) < 1$$. This area is crucial in number theory, especially concerning the behavior of analytic functions like the Riemann zeta function and Dirichlet L-functions, as it contains important information about their zeros and convergence properties.
Dirichlet series: A Dirichlet series is a type of infinite series of the form $$D(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $a_n$ are complex coefficients and $s$ is a complex variable. These series are a powerful tool in analytic number theory, linking properties of numbers with functions, especially through their relationship with zeta functions and multiplicative functions.
Euler Product Formula: The Euler Product Formula expresses the Riemann zeta function as an infinite product over all prime numbers, highlighting the deep connection between prime numbers and the distribution of integers. This formula shows that the zeta function can be represented as $$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$ for Re(s) > 1, linking analytic properties of the zeta function to number theory through primes.
Functional Equation: A functional equation is a mathematical equation that specifies a relationship between the values of a function at different points. These equations often arise in number theory and analysis, linking various properties of functions like the Riemann zeta function or Dirichlet series, helping us understand their behavior across different domains.
Logarithmic Derivative: The logarithmic derivative of a function is defined as the derivative of the logarithm of that function, providing useful information about growth rates and multiplicative structures. This concept is particularly relevant in number theory, as it helps analyze functions like the Riemann zeta function and Euler's totient function, especially in contexts involving primes and arithmetic progressions.
Non-vanishing: Non-vanishing refers to a property of certain mathematical functions where the function does not equal zero at specific points within a given domain. In the context of analytic number theory, especially regarding the Riemann zeta function, non-vanishing is critical for understanding its role in the distribution of prime numbers and is linked to various properties of the zeta function and its zeros. This concept plays a significant role in analyzing the Riemann Hypothesis and the Prime Number Theorem (PNT).
Poles: In the context of analytic number theory, poles refer to certain points in the complex plane where a meromorphic function, such as the Riemann zeta function, becomes undefined or diverges to infinity. The behavior of a function near its poles is crucial for understanding its properties, including analytic continuation and non-vanishing values on specific lines in the complex plane.
Prime Number Theorem: The Prime Number Theorem describes the asymptotic distribution of prime numbers, stating that the number of primes less than a given number $n$ is approximately $\frac{n}{\log(n)}$. This theorem establishes a connection between primes and logarithmic functions, which has far-reaching implications in analytic number theory, especially in understanding the distribution of primes and their density among integers.
Riemann Hypothesis: The Riemann Hypothesis is a conjecture in number theory that states all non-trivial zeros of the Riemann zeta function lie on the critical line in the complex plane, where the real part of s is 1/2. This hypothesis is crucial as it connects the distribution of prime numbers to the properties of analytic functions, influencing various aspects of number theory and its applications.
Riemann zeta function: The Riemann zeta function is a complex function defined for complex numbers, which plays a pivotal role in number theory, particularly in understanding the distribution of prime numbers. It is intimately connected to various aspects of analytic number theory, including the functional equation, Dirichlet series, and the famous Riemann Hypothesis that conjectures all non-trivial zeros of the function lie on the critical line in the complex plane.
Trivial zeros: Trivial zeros refer to the specific zeros of the Riemann zeta function, which occur at negative even integers: $$s = -2, -4, -6, \ldots$$ These zeros are significant in understanding the overall behavior of the zeta function and have implications for the distribution of prime numbers, as well as connections to the functional equation and the non-vanishing of the zeta function at certain critical lines.