Riemann Zeta Function: Convergence (Part 1) | Complex Analysis Tutorial

This lecture covers fundamental concepts of infinite series convergence, including the Cauchy criterion for series convergence (a series converges iff its partial sums form a Cauchy sequence), the relationship between absolute convergence and convergence (every absolutely convergent series converges), and the comparison test for series of non-negative terms. The lecture also presents the root test for series convergence, which states that for a series of non-negative terms, if the limit superior of the nth root of the terms is less than 1, the series converges; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.

This lecture covers fundamental concepts of infinite series in Real Analysis, including partial sums, Cauchy criterion, and various convergence tests: p-series test (converges for p > 1), comparison test, limit comparison test, ratio test, root test, Raabe's test, logarithmic test, telescoping series, Leibniz test for alternating series, and the distinction between absolute and conditional convergence. The instructor demonstrates these concepts through detailed examples and practice problems.

This section develops the theory of infinite series and their convergence. Given sequence x_n, the series Σx_n converges to L if the partial sums S_K = Σx_n (n=1 to K) converge to L as K → ∞. A necessary condition for convergence is lim(x_K) = 0, though this is not sufficient (harmonic series Σ1/n diverges despite terms approaching 0). Properties of convergent series: Σ(x_n ± y_n) converges to Σx_n ± Σy_n, and Σ(αx_n) converges to αΣx_n. Geometric series Σr^n converges to r/(1-r) if |r| < 1. The Comparison Test: if |x_n| ≤ y_n and Σy_n converges, then Σx_n converges absolutely; if x_n ≥ y_n ≥ 0 and Σy_n diverges, then Σx_n diverges. The Cauchy Condensation Test: for decreasing non-negative x_n, Σx_n converges iff Σ2^n x_{2^n} converges. This gives: Σ1/n^p converges if p > 1, diverges if p ≤ 1; Σ1/(n^p log n) converges if p > 1, diverges if p ≤ 1. Examples: Σsin(n²)/n² converges (|sin(n²)/n²| ≤ 1/n²), while Σlog(n)/n diverges (log(n)/n ≥ 1/n for large n).

This segment introduces absolute convergence, where a series converges if the series of absolute values converges. The instructor explains that absolute convergence implies the original series also converges. The Comparison Test is introduced as a method to determine convergence by comparing a series to another with known behavior. For positive-term series, if a series is smaller than a convergent series, it converges; if larger than a divergent series, it diverges.

This comprehensive section establishes the foundational concepts of infinite series. An infinite series is constructed from a sequence of real numbers by summing terms: x1 + x2 + x3 + ... + xn + ... Partial sums (Sk) are calculated by summing terms up to the k-th term. Convergence occurs when the sequence of partial sums has a limit, which becomes the series sum. Divergence occurs when no limit exists. The Nth Term Test states that if lim(n→∞) xn ≠ 0, the series diverges, though the converse is not true. The Cauchy Criterion provides a powerful alternative: a series converges if and only if for every positive epsilon, there exists N such that for all m, n > N, |Sm - Sn| < epsilon. The Monotone Convergence Theorem states that for series with non-negative terms, convergence occurs if and only if partial sums are bounded above. The Comparison Test states that if 0 ≤ xn ≤ yn and Σyn converges, then Σxn converges; if Σyn diverges and xn ≥ yn ≥ 0, then Σxn diverges. The Limit Comparison Test states that if lim(n→∞) xn/yn = L where 0 < L < ∞, both series either converge or diverge together. These tests provide systematic approaches for analyzing series behavior.

Cauchy's Integral Test determines the convergence of an infinite series by comparing it to an improper integral: if f(x) is a positive, monotonic decreasing, integrable function defined for x ≥ 1, then the series ∑uₙ from n=1 to ∞ converges if and only if the improper integral ∫₁^∞ f(x)dx converges (has a finite value); if the integral diverges to infinity, the series also diverges.

Cauchy's Integral Test determines whether an infinite series converges or diverges by comparing it to an improper integral; the test can be applied when the series terms are positive and decreasing, and the series ∑aₙ converges if and only if the improper integral ∫₁^∞ f(x)dx converges, where f(n) = aₙ.

The Cauchy Integral Test states that for a series ∑f(n) where f(x) is positive, continuous, and decreasing for x ≥ 1, the series converges if and only if the improper integral ∫₁^∞ f(x) dx converges to a finite value; if the integral diverges to infinity, the series also diverges. This test provides a powerful method for determining the convergence behavior of infinite series by converting the problem into an integral evaluation.

Cauchy's Integral Test states that for a positive series, the series converges if and only if the corresponding improper integral converges. To apply this test, replace the discrete variable n with a continuous variable x, then evaluate the integral from the starting point to infinity. For the series with general term 1/(n(ln n)³), the integral becomes ∫dx/(x(ln x)³) from 2 to infinity. Using substitution t = ln x, the integral simplifies to ∫dt/t² from ln 2 to infinity, which evaluates to a finite value, indicating the series converges.

The Cauchy Integral Test states that for a positive, monotonically decreasing function f(x), the series ∑f(n) from n=m to infinity converges if and only if the improper integral ∫f(x)dx from m to infinity converges. For the p-series ∑1/n^p, the integral ∫x^(-p)dx from 1 to infinity converges if and only if p > 1. Thus, the p-series converges if p > 1 and diverges if p ≤ 1.

The Riemann zeta function ζ(s) is defined as the infinite series Σ(n=1 to ∞) of 1/n^s, where s is a complex number. This function is central to the study of prime numbers and is defined for complex numbers s with real part greater than 1. The function can be extended to other values of s through analytic continuation.

The Riemann zeta function ζ(s) is defined as the sum from k=1 to infinity of 1/k^s, which can be written as ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ... This function converges when the real part of s is greater than 1 and can be analytically continued to the entire complex plane except for a simple pole at s=1.

The Riemann Zeta function ζ(s) = ∑(n=1 to ∞) 1/n^s is a cornerstone of mathematics, converging for Re(s) > 1. Riemann extended it to the entire complex plane using analytic continuation, revealing trivial zeros at negative even integers and conjecturing all non-trivial zeros lie on the critical line Re(s)=1/2 (the Riemann Hypothesis). Euler connected it to primes via the product formula ζ(s) = ∏(p prime) 1/(1-p^(-s)). Riemann further developed an explicit formula expressing the prime-counting function π(x) as a sum involving Zeta zeros, where the main term x/ln(x) gives the Prime Number Theorem asymptotic, and correction terms correspond to each non-trivial zero. Landau reformulated the hypothesis probabilistically using the Liouville function λ(n), conjecturing ∑(k=1 to n) λ(k) = O(n^(1/2+ε)), connecting number theory to random walk theory.

The Riemann zeta function ζ(s) = ∑(n=1 to ∞) 1/n^s connects number theory to complex analysis. For Re(s) > 1, it converges and encodes prime distribution through the Euler product formula: ζ(s) = ∏(p prime) 1/(1 - 1/p^s). This product shows ζ(s) has no zeros in this region. To extend ζ(s) to the entire complex plane, mathematicians multiply by gamma factors to create the completed zeta function ξ(s), which can be expressed as the Fourier transform of a rapidly decaying theta function. The Riemann hypothesis states that all non-trivial zeros lie on the critical line Re(s) = 1/2, with zeros symmetric about both axes. The number of zeros up to height T grows like N(T) ≈ T log T, with average spacing ~1/log T. While numerical evidence supports the hypothesis, fine-scale distribution remains mysterious.

The Riemann zeta function ζ(s) is defined as the infinite series 1/1^s + 1/2^s + 1/3^s + ... where s is a complex number. When the real part of s (denoted Re(s)) is greater than 1, this series converges to a specific finite value. For other values of s, the function is extended using analytic continuation, which is a method of extending functions to regions where they were originally undefined while preserving differentiability. This extension is unique, meaning there is exactly one way to define the function for all complex numbers except s = 1, where it has a pole.