A non-trivial infinity of zeroes with real part one-half

[Loren Booda]

Other than the Riemann Zeta function, what equation has a non-trivial infinity of zeroes with real part one-half?

 

[ramsey2879]

Other than the Riemann Zeta function, what equation has a non-trivial infinity of zeroes with real part one-half?

T(a) = a*(a-1)/2
T(1/2 + b*i) has an imaginary part equal to zero

I(a) = a*(a-i)/2
I(b +(1/2)*i) also has imaginary part equal to zero

Let c = a+b
Then N(a+bi) =I(b+a +(a)i) - T(a+(b+a)i) has a zero when a = 1/2

 

[Office_Shredder]

What equation? How about Re(z) - 1/2 = 0

Maybe you're looking for holomorphic functions

 

[Dragonfall]

How about f(z)=0?

 

[Loren Booda]

Perhaps holomorphic functions agree with what is meant by "non-trivial" in the definition of the Riemann Zeta function. In this latter regard the above examples appear trivial.

Also, by "real part" I implied that there was an "imaginary part." Is the Riemann-equivalent "alternating Zeta function" (a real, generalized alternating harmonic series) holomorphic?

 

[Hurkyl]

Perhaps holomorphic functions agree with what is meant by "non-trivial" in the definition of the Riemann Zeta function.

No. "Trivial zero of the Riemann zeta function" is simply a synonym for "negative even integer". We choose to call them the "trivial" zeroes because it's very easy to prove they are zeroes, and they are irrelevant to the Riemann hypothesis.

(I highly doubt you have ever see the word "nontrivial" in any definition of the Riemann Zeta function)

Office_Shredder mentioned "holomorphic functions" because he was trying to guess what the heck you meant by "equation".

 

[camilus]

(I highly doubt you have ever see the word "nontrivial" in any definition of the Riemann Zeta function)

Of course, the RZF is just a simple function, the infinite series of th inverse intergers raised to the s power argument of the RZF. Nontrivial zeros are nowhere mentioned there, but they have to be mentioned in a definition of the RH, because the RH is just a conjecture about the RZF when the argument s now becomes a complex z = a + bi. Then Euler found the connection with the primes, which makes everything interesting. When we talk about the RH, you have to specify that you're talking about the nontrivial zeros.

 

[HallsofIvy]

I'm wondering what in the world a "trivial infinity" would be!

 

[Loren Booda]

Sorry, it should read "infinite number," not "infinity."

 

[Petek]

Other than the Riemann Zeta function, what equation has a non-trivial infinity of zeroes with real part one-half?

Perhaps this is what you're looking for:

Consider the function sin(z). This has zeroes at all values n*pi, where n is an integer. Now rotate by 90 deg to get sin(iz). This has zeroes at at all values n*pi*i, where again n is any integer. Finally, translate by 1/2, to get sin(i(z - (1/2))). This last function has zeroes at all values (1/2) + n*pi*i, which have real part 1/2, as desired.

 

[Loren Booda]

Thanks, Petek.

I had earlier in this thread misinterpreted "non-trivial" as representing the seeming "random" distribution of primes on the line Re(z)=1/2. The sine function you mention generates answers with much more simple functions. Despite my initial confusion, yours is an elegant answer, although not akin to the class of function I was looking for.

Perhaps sequences like prime numbers could be found through such functions as I suggest. Perhaps I'm barking up the wrong tree.

 

[Petek]

You might want to look at the Weierstrass factorization theorem, which allows one to construct functions that have zeroes at any sequence {a_n} that satisfies certain growth conditions.

 

[Loren Booda]

The Weierstrass factorization theorem may introduce the tools needed to find functions in the complex plane similar to that described in the Riemann hypothesis, and generate series of numbers relating unique properties to the harmonic series.

Weierstrass' elementary factors seem to enable this. I would not be surprised in Riemann used a form of this theorem if it were available to him. This concept is very close to what I had envisioned.

 

[Petek]

The Weierstrass factorization theorem may introduce the tools needed to find functions in the complex plane similar to that described in the Riemann hypothesis, and generate series of numbers relating unique properties to the harmonic series.

Weierstrass' elementary factors seem to enable this. I would not be surprised in Riemann used a form of this theorem if it were available to him. This concept is very close to what I had envisioned.

Yes, Riemann knew a product expansion of the zeta function in terms of its zeroes. It's now called the Hadamard product.

 

[zetafunction]

the question is that someone can proof that 'at leat the 40 % of zeros have real part 1/2' is there an argument to generalize this to 100 % , how can we really know that the 40 50 % of zeros have real part 1/2 if there is an infinite amount of them.

 

[Petek]

the question is that someone can proof that 'at leat the 40 % of zeros have real part 1/2' is there an argument to generalize this to 100 % , how can we really know that the 40 50 % of zeros have real part 1/2 if there is an infinite amount of them.

To reply to the second part of your post, define N(T) to be the number of zeros s + it of the zeta function such that 0 < s < 1 and 0 < t \leq T and define N_0(T) to be the number of zeros such that 0 < t \leq T. In other words, N(T) is the number of zeros in the critical strip with imaginary part > 0 and \leq T. N_0 is the number of zeros on the corresponding critical line. The Riemann Hypothesis asserts that N_0(T) = N(T). What has been proved is that

N_0 > \frac{2}{5}N(T)

for sufficiently large T. This is what's meant when saying that at least 40% of the (non-trivial) zeros have real part \frac{1}{2}.

Hope that helps.