February 5th, 2013, 07:08 AM  #1 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Zeta on the critical line
Prove that zeta has no zeros (Riemann's) on the critical line (real part 1/2) with an integral imaginary part, that is, the imaginary part is an integer. I am halfway through of proving it, would be appreciated if someone do it before me. 
February 5th, 2013, 12:20 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,705 Thanks: 670  Re: Zeta on the critical line
If you do it and the proof is correct, you will become rich (prize is $1,000,000.) and famous.

February 5th, 2013, 01:41 PM  #3 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  Re: Zeta on the critical line
Mathman I am not sure why you say so. This statement is not the Riemann Hypothesis at all. I like very much things like "I am half way through the derivation"..If you don't have yet the full derivation, how do you know you are half the way through it ?? Having said that, I don't have the time to work on this now, but I would be glad to see later how it is proven. Thanks! 
February 5th, 2013, 05:57 PM  #4  
Math Team Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions  Re: Zeta on the critical line Quote:
 
February 5th, 2013, 09:47 PM  #5  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Zeta on the critical line Quote:
Quote:
 
February 6th, 2013, 05:34 AM  #6 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Zeta on the critical line
Okay, a sketch of my line of proof : If has no zeros at for positive integer n, then to prove it, we need a sum with denominator . After one or two research, I found this one : I am almost sure that this sum converges but not able to prove it. I've also checked with PARI and it returned : But I do not trust computerized results and want to prove it by hands, anyone interested? 
February 6th, 2013, 11:13 AM  #7 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  Re: Zeta on the critical line
Hi Balarka, That 's probably going to be tough to find an analytical value for this sum! Also constitutionally, I think that does not mean anything. I don't know what is PARI but it has to calculate until a certain N, right?, So what it proves is that there is no root of this form for n<N, but you can never check until infinity like this. The sum can converge (because the derivative of zeta will tend to zero enough fast) until one of the terms blows up if there is a zero of that form. Do you agree? 
February 6th, 2013, 08:32 PM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Zeta on the critical line Quote:
So, what about it? Can this be proved that zeta has no nontrivial zeros with imaginary part an integer or is it a conjecture ? atleast I never heard of it. EDIT: Can we use to bound up the sum?  
February 6th, 2013, 09:05 PM  #9 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  Re: Zeta on the critical line
Hi, That's funny I wanted to write "computationally" and the spellcheck change me this to "constitutionally"!! Anyway, I don't know about it. Do you have a reason in the first place for assuming there can not be zeros of that form? I personally don't have an argument for that. I think the last formula you wrote will not help you. My guess is that it is Balarka's conjecture! 
February 6th, 2013, 09:39 PM  #10  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Zeta on the critical line Quote:
Quote:
 

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