what did tchbeshevy contribute to the riemann hypothesis

The Riemann Hypothesis

Summary:

When studying the distribution of prime numbers Riemann extended Euler'southward zeta function (divers just for southward with existent part greater than one) to the unabridged complex plane (sans unproblematic pole at s = 1).  Riemann noted that his zeta function had trivial zeros at -two, -four, -vi, ...; that all nontrivial zeros were symmetric about the line Re(s) = 1/ii; 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 permit us to greatly acuminate many number theoretical results.  For case, in 1901 von Koch showed that the Riemann hypothesis is equivalent to:

Just it would non make factoring any easier!  In that location are a couple standard ways to generalize the Riemann hypothesis.

1. The Riemann Hypothesis:

Euler studied the sum

for integers due south>1 (conspicuously(1) is infinite).  Euler discovered a formula relating(2chiliad) to the Bernoulli numbers yielding results such equally and.  But what has this got to exercise with the primes?  The answer is in the following production taken over the primes p (as well discovered by Euler):

Euler wrote this as

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

where the gamma function(s) is the well-known extension of the factorial function ((north+i) = 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 circuitous numbers due south.)

The Riemann zeta function has the trivial zeros at -2, -4, -6, ... (the poles of(s/two)).  Using the Euler product (with the functional equation) information technology is easy to show that all the other zeros are in the critical strip of non-real circuitous numbers with 0 < Re(s) < 1, and that they are symmetric most the critical line Re(southward)=1/2.  The unproved Riemann hypothesis is 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 twoscore% of the zeros in the critical strip are on the critical line [Conrey89].  All the same, there is notwithstanding a chance that the Riemann hypothesis is false.  From Baronial of 2001 through 2005, Sebastian Wedeniwski ran ZetaGrid which verified that the first 100 billion zeros were on the disquisitional line.

2. Who cares?

In 1900 Hilbert listed proving or disproving this hypothesis as ane of the most important unsolved problems confronting modern mathematics and it is central to understanding the overall distribution of the primes.  When Hadamard and de la Vallee Poussin proved the prime number number theorem, they really showed

for some positive constant a, and they did this by bounding the real part of the zeros in the critical strip away from 0 and 1.  The fault term is directly dependent on what was known nearly the nada-free region inside the critical strip.  As our knowledge of the size of this region increases, the error term decreases.  In fact, in 1901 von Koch showed that the Riemann hypothesis is equivalent to

There are many results like this, see, for example [BS96].

Generalizations of RH

Retrieve once again our starting point from Euler:

Why should the numerators all be one?  One important mode to modify the series is to replace the numerators with functions χ(n) called Dirichlet characters (these can exist viewed as functions for which there exists a positive integer 1000 with χ(north + k) = χ(n) for all n, and with χ(n) = 0 whenever gcd(due north, thousand) > i).  The resulting infinite sum L(?,s) is a Dirichlet L-function.  Once again we analytically keep the function to one that is meromophic on the entire complex airplane.   The extended Riemann Hypothesis is that for every Dirichlet character χ and the zeros L(χ,s) = 0 with 0 < Re(s) < one, have real role ane/2.  The distributions of the zeros of these L-functions are closely related to the number of primes in arithmetics progressions with a fixed deviation grand.  Should the extended Riemann Hypothesis be proven, so Miller'due south exam would provide an efficient primality proof for general numbers.  Come across, for example, [BS96 viii.5-6].

Another mode to generalize Euler's sum is to leave the field of rational numbers, and replace the denominators with the norms of the non-zero ideals (special sets of elements) in a finite field extention of the rationals K (called a number field).  The resulting sum is the Dedekind zeta-function of K and can over again exist analytically continued.  These zeta functions also accept a simple pole at zero and infinitely many zippo in the critical region.  The generalized Riemann Hypothesis is again that the zeros in the critical region all accept real role i/2.  Come across, for example, [BS96 viii.7].

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.

gladdenachim1992.blogspot.com

Source: https://primes.utm.edu/notes/rh.html

Share this post