July 16th, 2013, 01:19 PM  #1  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Understanding the Riemann hypothesis Quote:
 
July 16th, 2013, 01:45 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Math Q&A Part 8 Quote:
for all x >= 2657. Or to make it even easier, see http://arxiv.org/abs/math/0008177 You don't have to understand the zeta function at all to understand RH. (Smart money says you'll have to understand it to follow the proof of RH... but we don't have one of those yet.)  
July 16th, 2013, 03:01 PM  #3  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Math Q&A Part 8 Quote:
 
July 17th, 2013, 04:53 AM  #4 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Understanding the Riemann hypothesis
I'd rather like to help one understand the Riemann Hypothesis than a consequence of it. It takes a lot to prove that the bound on Pi(x)  Li(x) is equivalent to Riemann Hypothesis. We have a function which is convergent only for Re(z) > 1. There are certain analytical methods which gives you some other function that agrees with the former one on Re(z) > 1 and also have it's own convergent values on some other domain. We can analytically continue zeta this way through the complex plane. RH says that all the zeros of zeta, except some which lies on the negative even numbers, are situated on Im(z) = 1/2. Balarka . 
July 17th, 2013, 05:25 AM  #5 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  Re: Understanding the Riemann hypothesis
mathbalarka meant Re(z)=1/2

July 17th, 2013, 06:23 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Understanding the Riemann hypothesis Quote:
'Of course' the two are logically equivalent. Quote:
 
July 17th, 2013, 07:18 AM  #7  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Understanding the Riemann hypothesis Quote:
Quote:
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July 17th, 2013, 07:32 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Understanding the Riemann hypothesis Quote:
Quote:
For example, in the generality you present it, there's nothing special about a continuation  there are beth_2 extensions of zeta to the values with Re(z) <= 1. It does make sense to talk about "the" analytic continuation of zeta, but only because of a manifestly nonobvious theorem.  
July 17th, 2013, 07:40 AM  #9  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Understanding the Riemann hypothesis Quote:
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July 17th, 2013, 07:50 AM  #10  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Understanding the Riemann hypothesis Quote:
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