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Analytic Number Theory: Dirichlet series - Oxford Mathematics 4th Year Student Lecture

Oxford Mathematics・30 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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Summary

00:00

Analyzing Sequences with Der Series and Zeta

  • Multiplicative functions in Du convolution were discussed in the last lecture, emphasizing understanding behavior on prime powers to determine overall behavior.
  • Generating functions were introduced as a way to comprehend sequences through the sum of a_n*x^n, allowing recovery of sequence terms from the generating function.
  • The indicator function of primes was suggested as a potential generating function to understand prime distribution.
  • Der series, involving the sum of a_n*n^s or n^(-s), were presented as a more effective way to analyze sequences.
  • A basic lemma was provided to ensure convergence of the der series to an analytic function for the real part of s greater than one.
  • The Riemann zeta function, denoted as C(s), was highlighted as a significant example of a der series.
  • Analytic continuation of the Riemann zeta function was proven, extending its definition to the region where the real part of s is greater than -2.
  • A complex formula for zeta(s) was derived, showcasing the extension of the function's definition.
  • The proof of analytic continuation involved partial summation and integration by parts to establish the validity of the extended formula.
  • The process of integrating the fractional part of T over t^s+1 was detailed, leading to the complex formula for zeta(s) in a broader range of s values.

26:00

Properties of Riemann Zeta Function Analyzed

  • The integral of the fractional part of U minus a half over any interval of length one is zero.
  • The main term contribution is adjusted by subtracting one from the integral to maintain a specific property.
  • The function G of T is equal to the function G of the fractional part of T.
  • The definition can be changed to the integral between zero and T without altering the outcome.
  • The overall integral is expressed as 12 minus s * s + 1 times the integral from 1 to Infinity of a specific function.
  • The complex analytic function makes sense when the real part of s is greater than minus one.
  • Analytic continuation is proven in a unique way, showcasing the special properties of the Riemann zeta function.
  • The Riemann zeta function evaluated at specific points yields distinct results, such as -1 equating to -1/12.
  • The function Zeta of s has a polynomial growth when the real part of s is greater than -19/10.
  • 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.

51:25

Analytic Continuation for Zeta Function Estimation

  • The proof involves the sum of 1/n multiplied by s, starting from n=1 to n, and then continuing from n=n+1 to infinity, with a focus on the real part of s being greater than one. By utilizing partial summation, the conclusion is reached that the integral from n to infinity of the fractional part of T over t^s+1 DT is bounded by specific values, leading to an approximate formula for the zeta function.
  • Through an alternative analytic continuation of the Riemann zeta function, a method is presented where an integer parameter, capital N, can be chosen to approximate the zeta function effectively even when the full series diverges. This approach, based on partial summation arguments, allows for a precise estimation of the zeta function within a wider region, proving to be a valuable tool despite the original series' divergence.
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