Has anything similar to the Riemann hypothesis ever solved

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The discussion centers on the Riemann hypothesis and its parallels with other mathematical conjectures, particularly the Weil conjectures. Participants explore whether any functions have been proven to have their nontrivial zeros consistently occurring at specific values. Examples such as the sine and cosine functions, which have zeros on the real axis, and the function f(x)=e^x-1, which has zeros on the imaginary axis, are cited. The conversation emphasizes the complexity of proving such properties for various functions.

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mustang19
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Has anything similar to the Riemann hypothesis ever been solved?

Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
 
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mustang19 said:
Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
It is not the function that takes particular values, the (nontrivial) zeros of the function (probably) occur at particular values.

It is easy to find some other examples, e.g. the zeros of the sine and cosine function are all on the real axis, the zeros of f(x)=ex-1 are all on the imaginary axis, and so on.
I don't know if there has been a function where such a proof has been found after considerable effort, but I would expect that there are hard examples.
 
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