--%>
Assignment Help - homework help

State Prime number theorem

Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often difficult. We will introduce one fundamental and extremely significant theorem about the distribution of prime numbers. (Its proof is one of the difficult ones!)

Let x be any positive number. We denote by Π (x) the number of prime numbers less than or equal to x. The prime number theorem was conjectured by Gauss in the year 1791 (at the age of 14!), but was not proved until 1896, when it was proved independently through Jacques Hadamard and Charles de la Vallee Poussin.

982_prime number.jpg

(Remember that ln x denotes the natural logarithm of x: ln x = loge x.)

   Related Questions in Mathematics

  • Q : Abstract Algebra let a, b, c, d be

    let a, b, c, d be integers. Prove the following statements: (a) if a|b and b|c. (b) if a|b and ac|bd. (c) if d|a and d|b then d|(xa+yb) for any x, y EZ

    		•
  • Q : What is the definition of a group Group

    Group: Let G be a set. When we say that o is a binary operation on G, we mean that o is a function from GxG into G. Informally, o takes pairs of elements of G as input and produces single elements of G as output. Examples are the operations + and x of

    		•
  • Q : Who independently developed

    Who independently developed a model for simply pricing risky assets?

    		•
  • Q : Problem on budgeted cash collections

    XYZ Company collects 20% of a month's sales in the month of sale, 70% in the month following sale, and 5% in the second month following sale. The remainder is not collectible. Budgeted sales for the subsequent four months are:     

    		•
  • Q : Theorem-G satis es the right and left

    Let G be a group. (i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b. (ii) If g ≡ G, then (g-1)

    		•
  • Q : Examples of groups Examples of groups:

    Examples of groups: We now start to survey a wide range of examples of groups (labelled by (A), (B), (C), . . . ). Most of these come from number theory. In all cases, the group axioms should be checked. This is easy for almost all of the examples, an

    		•
  • Q : Relationships Between Data Introduction

    Relationships Between Data - Introduction to Linear Regression Simple Regression Notes If you need guidance in terms of using Excel to run regressions, check pages 1 - 10 of the Excel - Linear Regression Tutorial posted to th

    		•
  • Q : Statistics math Detailed explanation of

    Detailed explanation of requirements for Part C-1 The assignment states the following requirement for Part 1, which is due at the end of Week 4: “Choose a topic from your field of study. Keep in mind you will need to collect at least [sic] 3- points of data for this project. Construct the sheet y

    		•
  • Q : Properties of a group How can we say

    How can we say that the pair (G, o) is a group. Explain the properties which proof it.

    		•
  • Q : Profit-loss based problems A leather

    A leather wholesaler supplies leather to shoe companies. The manufacturing quantity requirements of leather differ depending upon the amount of leather ordered by the shoe companies to him. Due to the volatility in orders, he is unable to precisely predict what will b

    		•