Tìm kiếm
Bí mật
(To be updated)
What is Analytic Number Theory (ANT) ?
- Number Theory is concerned with the properties of Z and N, e.g. "Every positive integer is the sum of four squares".
- Some results are best tackled by using analysis, these problems often involve approximations, e.g. "roughly" how many prime numbers are there below x (Prime Number Theorem).
- But sometimes there is no obvious reason why real or complex analysis comes to the stage.
----------------
Examples of elegant results that are stated in a discrete / algebraic manner but can be proved using analytic methods:
Every positive integer can be written as a sum of 4 sqaures, or 9 cubes, or 19 4th powers, etc. [proved 1986, answering a problem of Waring, 1770]
The sequence of promes contains arbitrarily long arithmetic progressions (e.g. 7, 37, 67, 97, 127, 157 has length 6) [proved 2004]
The ring {a + b \sqrt{14}} is a Euclidean Domain [proved 2004]
Every number beyond 10^1347 is a sum of 3 primes (Goldbach, in 1742, asked if every odd number at least 7 is a sum of 3 primes --> lots of numerical checking are left)
----------------
Many hallmarks of "elementary" analytic number theory are concerned with arithmetic functions (to be defined later) and prime numbers.
----------------
A first big question that we shall address is about how fast \pi(x) grows with x, where \pi(x) := # {p \leq x : p is prime}.
Euclid's Theorem: There are infinitely many primes.
Proof: elementary.
Some numerical calculation:
\pi (10^8) = 5,761,455
\pi (10^12) = 37,607,912,018
\pi (10^16) = 279,238,341,033,925
With these 10th powers:
\pi(x) / x = 0.057... , 0.037... , 0.027... (seems to decrease, but how fast ?)
1 / log(x) = 0.054..., 0.036... , 0.027...
Question: is it true that \pi(x) / x ~ log(x) , that is, \pi(x) ~ x / log(x) ?
Prime Number Theorem [Hadamard, de la Vallee Poussin, 1896]:
\pi(x) ~ x / log(x) as x --> \inf (infinity)
(Definition: f(x) ~ g(x) with g(x)>0, iff(def) f(x)/g(x) --> 1 as x --> \inf)
----------------
Tables show that the detailed distribution of primes is very erratic. Lots of pairs of primes differ by 2.
Open Problem: Are there infinitely many "Prime Twins" p, p+2 (i.e. p and p+2 are both primes) ?
There are also large gaps between the primes:
e.g. N! + 2 , N! + 3, ... , N! + N are all composite numbers for N > 1.
Open Problem (stated in a vague way): Roughly how large is the largest gap between consecutive primes p \leq N ?
- Hung Phu Phan yêu thích
