Lectures # 5 and 6 : The Prime Number Theorem . Noah Snyder
نویسنده
چکیده
Riemann used his analytically continued ζ-function to sketch an argument which would give an actual formula for π(x) and suggest how to prove the prime number theorem. This argument is highly unrigorous at points, but it is crucial to understanding the development of the rest of the theory. Notice that log ζ(s) = ∑ p ∑ n 1 np −ns for Re(s) > 1. Letting J(x) = ∑ pk≤x 1 k , notice that log ζ(s) = ∫∞ 0 x−sdJ(x) again for Re(s) > 1. Now use integration by parts to get
منابع مشابه
Lectures on Analytic Number Theory
1. What is Analytic Number Theory? 2 1.1. Generating functions 2 1.2. Operations on series 3 1.3. Some interesting series 5 2. The Zeta Function 6 2.1. Some elementary number theory 6 2.2. The infinitude of primes 7 2.3. Infinite products 7 2.4. The zeta function and Euler product 7 2.5. Infinitude of primes of the form 4k + 1 8 3. Dirichlet characters and L functions 9 3.1. Dirichlet character...
متن کاملLectures on Integer Matrices
Introduction 2 Lecture 1. Hermite and Smith normal forms 3 Lecture 2. Integral similarity and the Latimer-MacDuffee-Taussky theorem 8 Lecture 3. Ideal class numbers, integral matrices nonderogatory modulo every prime and maximal orders of number fields 12 Lecture 4. Factorizations of integer matrices as products of elementary matrices, involutions etc 20 Lecture 5. Additive commutators. Solving...
متن کاملThick subcategories of perfect complexes over a commutative ring
0→ Ps → · · · → Pi → 0 where each Pi is a finitely generated projective R-module. Let P the full subcategory of D consisting of complexes isomorphic to perfect complexes. These are precisely the compact objects, also called small objects, in D. These notes are an abstract of two lectures I gave at the workshop. The main goal of the lectures was to present various proofs of a theorem of Hopkins ...
متن کاملLectures on Local Cohomology
This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Gr...
متن کاملLectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms
Notice that this means if we change variables ( x′ y′ ) = v′ = Av (the entries of A are integers) then Q(x′, y′) = (Av) QAv = v (A QA)v. Therefore, if Q represents a number then so does A QA. In particular, if A has an inverse with integer entries, then we get that Q and A QA represent all the same numbers. Clearly if A has an inverse B with integer entries, then det A detB = det AB = 1, thus d...
متن کامل