Интересные факты

· Знаменит ответ Гильберта на вопрос о том, каковы будут его действия, если он по какой-либо причине проспит пятьсот лет и вдруг проснётся. Математик ответил, что первым делом он спросит, была ли доказана гипотеза Римана.

· Гипотеза Римана относится к знаменитым открытым проблемам математики, в число которых в своё время входила и теорема Ферма. Как известно,Ферма сделал запись о том, что доказал свою теорему, не оставив самого доказательства, и тем самым бросил вызов следующим поколениям математиков. Британский математик Г. Х. Харди использовал ситуацию с этими проблемами для обеспечения собственной безопасности во время морских путешествий. Каждый раз перед отправкой в путешествие он отправлял одному из своих коллег телеграмму: ДОКАЗАЛ ГИПОТЕЗУ РИМАНА ТЧК ПОДРОБНОСТИ ПО ВОЗВРАЩЕНИИ ТЧК. Харди считал, что бог не допустит повторения ситуации с теоремой Ферма и позволит ему благополучно вернуться из плавания.^[18]

Summary:

When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one)

to the entire complex plane (sans simple pole at s = 1). Riemann noted that his zeta function had trivial zeros at -2, -4, -6, ...; that all nontrivial zeros were symmetric about the line Re(s) = 1/2; and that the few he calculated were on that line. The Riemann hypothesis is that all nontrivial zeros are on this line. Proving the Riemann Hypothesis would allow us to greatly sharpen many number theoretical results. For example, in 1901 von Koch showed that the Riemann hypothesis is equivalent to:

But it would not make factoring any easier! There are a couple standard ways to generalize the Riemann hypothesis..

1. The Riemann Hypothesis:

Euler studied the sum

for integers s>1 (clearly (1) is infinite). Euler discovered a formula relating (2k) to the Bernoulli numbers yielding results such as and . But what has this got to do with the primes? The answer is in the following product taken over the primes p (also discovered by Euler):

Euler wrote this as

Riemann later extended the definition of (s) to all complex numbers s (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of the Riemann zeta function:

where the gamma function (s) is the well-known extension of the factorial function ( (n+1) = n! for non-negative integers n):

(Here the integral form holds if the real part of s is greater than one, and the product form holds for all complex numbers s.)

The Riemann zeta function has the trivial zeros at -2, -4, -6, ... (the poles of (s/2)). Using the Euler product (with the functional equation) it is easy to show that all the other zeros are in the critical strip of non-real complex numbers with 0 < Re(s) < 1, and that they are symmetric about the critical line Re(s)=1/2. The unproved Riemann hypothesisis that all of the nontrivial zeros are actually on the critical line.

In 1986 it was shown that the first 1,500,000,001 nontrivial zeros of the Riemann zeta function do indeed have real part one-half [VTW86]. Hardy proved in 1915 that an infinite number of the zeros do occur on the critical line and in 1989 Conrey showed that over 40% of the zeros in the critical strip are on the critical line [Conrey89]. However, there is still a chance that the Riemann hypothesis is false. From August of 2001 through 2005, Sebastian Wedeniwski ran ZetaGrid which verified that the first 100 billion zeros were on the critical line.