Semjase

This proof proves Riemann's hypothesis wrong, I've posted it elsewhere but I never got an answer whats wrong with this proof.
Maybe someone could shed some light on this for the benefit of my friend Semjase.

From Euler's equation e^(i*pi) = -1

Let 0 + 0i = 1/1^s + 1/2^s + 1/3^s + 1/4^s . . .

Moving 1/2^s to the left side of the equation you get

-1/2^s = 1/1^s + 1/3^s + 1/4^s . . .

substituting -1 = e^i*pi

e^(i*pi)/2^s = 1/1^s + 1/3^s + 1/4^s . . .

add 1/2^s to both sides of the equation

1/2^s + e^(i*pi)/2^s = 1/1^s + 1/2^s + 1/3^s + 1/4^s . . .

which gives

0 + 0i = 1/2^s + e^(i*pi)/2^s

multiply both sides 2^(2*s)

0 + 0i = 2^s + (2^s)*e^(pi*i)

next

0 + 0i = (2^a)*2^(q*i) + (2^a)*2^(q*i)*e^(pi*i)

let 2^q = e^pi

0 + 0i = (2^a)*e^(pi*i) + (2^a)*e^(pi*i)*e^(pi*i)

let e^(pi*i) =-1

0 + 0i = (2^a)*-1 + (2^a)*-1*-1

0 + 0i = - 2^a + 2^a

Now (a) can be any real number therefore Riemann's hypothesis is incorrect

HallsofIvy

This makes no sense at all. Do you mean "if" that

Moving 1/2^s to the left side of the equation you get

-1/2^s = 1/1^s + 1/3^s + 1/4^s . . .

substituting -1 = e^i*pi

e^(i*pi)/2^s = 1/1^s + 1/3^s + 1/4^s . . .

add 1/2^s to both sides of the equation

1/2^s + e^(i*pi)/2^s = 1/1^s + 1/2^s + 1/3^s + 1/4^s . . .

which gives

0 + 0i = 1/2^s + e^(i*pi)/2^s

multiply both sides 2^(2*s)

0 + 0i = 2^s + (2^s)*e^(pi*i)

next

0 + 0i = (2^a)*2^(q*i) + (2^a)*2^(q*i)*e^(pi*i)

let 2^q = e^pi

0 + 0i = (2^a)*e^(pi*i) + (2^a)*e^(pi*i)*e^(pi*i)

let e^(pi*i) =-1

0 + 0i = (2^a)*-1 + (2^a)*-1*-1

0 + 0i = - 2^a + 2^a

Now (a) can be any real number therefore Riemann's hypothesis is incorrect[/quote]
That's non-sense. What you have proved is that "0= 0". Do you not understand that if you manipulate an equation, multiplying both sides of an equation by something involving the variable "x", you can introduce "new" solutions, getting an equation that has solutions the original equation didn't? For example to "disprove" the statement "The only solution to x- 1= 0 is 1" do the following: mutiply both sides of the equation by x to get . That factors as showing that both 1 and 0 satisfy the equation. You have simply manipulated the original equation to get a new equation that has all real numbers as solution. That says nothing about what numbers satisfy the original equation.

Dougy

Yes and in any case your original equation is not even the expression of Zeta in the strip ]0,1[, it is only valid for Re(s)>1 and it is known that Zeta does not have zero in that region...

mathbalarka

This is not the riemann zeta. The formal definition of riemann zeta is the functional equation :

.

The definition you wrote,

is only valid for .

But riemann conjectured that all the zeros of zeta lies in the region . And so, your proof isn't valid for this case since 1/2<1

greg1313

This appears to define the zeta in terms of itself.

Dougy

This is the functional equation of Zeta, showing a symmetry of the function with respect to the critical line Re(s)=1/2. I don't know if it is the formal definition, but it I think it defines Zeta uniquely.

mathbalarka

Yes. Every functional equation defines the function used in the equation uniquely (maybe) in terms of that function.

This is very important in calculating the values of for Re[s] < 1.

Let us calculate using this. We will get :



we know that ;





See, how useful it is?

.

CRGreathouse

Of course the functional equation only tells you how to find one side when you already know how to find the other. The usual way, I think, is to define it for Re(s) > 1 and then use the analytic continuation to define it for the rest of

Dougy

Yes Mathbalarka, that is what Ramanujan wrote in his letter to Hardy, namely . They thought this guy was crazy, but actually he knew this property of Zeta but did not know the right notation. I am not sure if he knew the Riemann paper or if he comes to it by himself..knowing how great of a mathematician he was, that would not surprised me!

agentredlum

If Dougy is right, that

why do we accept this result? To me it doesn't make sense, the sum on the left is not even close to the value on the right.

Can somebody explain the underlying philosophy that would force us to conclude such an 'absurdity', IMHO