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4. The Reimann zeta-function is defined by II. and is used in number theory to study the distribution of primes. (a) For values of x does ¢ converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 n=1 $(2) = 1 -£-/-/- n=1 1 (n + 2)² 4 3 (³)* = π² ∞ 1 n² 6 ζ(x) = Σ Use Euler's results to find the sum of the following series: ∞ ∞ 1. Σ 1 III. Σ n=3 (n − 2)4 - n=5 n=j nữ ∞ IV. Σ 1 ∞ ζ(4) = Σ n=1 n=2 n² 1 TA 90 n4 =

4. The Reimann zeta-function is defined by II. and is used in number theory to study the distribution of primes. (a) For values of x does ¢ converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 n=1 $(2) = 1 -£-/-/- n=1 1 (n + 2)² 4 3 (³)* = π² ∞ 1 n² 6 ζ(x) = Σ Use Euler's results to find the sum of the following series: ∞ ∞ 1. Σ 1 III. Σ n=3 (n − 2)4 - n=5 n=j nữ ∞ IV. Σ 1 ∞ ζ(4) = Σ n=1 n=2 n² 1 TA 90 n4 =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. The Reimann zeta-function ζ is defined by and is used in number theory to study the distribution of primes. (a) For values of x does ( converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 0 1. Σ n=3 Ç(2) = 1 (n + 2)2 II. 1. Σ(3) n=1 00 4 n=1 π Use Euler's results to find the sum of the following series: 1 (n - 2)4 00 ς(α) 4(x) = Ση n=1 2 0 III. Σ η=5 IV. Σ n=2 $(4) = 1 η2 00 n=1 = .4 η4 90
Transcribed Image Text:4. The Reimann zeta-function ζ is defined by and is used in number theory to study the distribution of primes. (a) For values of x does ( converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 0 1. Σ n=3 Ç(2) = 1 (n + 2)2 II. 1. Σ(3) n=1 00 4 n=1 π Use Euler's results to find the sum of the following series: 1 (n - 2)4 00 ς(α) 4(x) = Ση n=1 2 0 III. Σ η=5 IV. Σ n=2 $(4) = 1 η2 00 n=1 = .4 η4 90
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