Analytic Number Theory: Dirichlet series - Oxford Mathematics 4th Year Student Lecture
Oxford Mathematics・2 minutes read
The lecture discussed multiplicative functions in Du convolution, emphasizing prime powers behavior, and introducing generating functions to understand sequences through a_n*x^n sums. The Riemann zeta function was proven to have analytic continuation with a complex formula derived for zeta(s), showcasing unique properties and distinct results at specific points.
Insights
- Multiplicative functions in Du convolution are essential to understanding behavior on prime powers, which is crucial for determining overall behavior in sequences.
- The Riemann zeta function, a significant example of a der series, undergoes analytic continuation to extend its definition to the region where the real part of s is greater than -2, showcasing special properties and unique results like -1 equating to -1/12.
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Recent questions
What are multiplicative functions in Du convolution?
Multiplicative functions in Du convolution involve understanding behavior on prime powers to determine overall behavior.
How are generating functions used to comprehend sequences?
Generating functions are utilized to comprehend sequences through the sum of a_n*x^n, allowing recovery of sequence terms from the generating function.
What is the indicator function of primes?
The indicator function of primes is suggested as a potential generating function to understand prime distribution.
How is the Riemann zeta function denoted?
The Riemann zeta function, denoted as C(s), is a significant example of a der series.
What is the derivative of the Riemann zeta function?
The derivative of the Riemann zeta function is calculated as minus the sum of log n over n to the power of s for s greater than one.
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