On Jun 1, 2010, at 8:13 AM, Anne Driver wrote:
Hello,
I am new to this list, and relatively new to Sage. I'm puzzled by
the logic of one part of Sage though.
Although I don't have access to Mathematica at the minute on this
computer, I know if I compute the first zero, I get something like
In[1] = ZetaZero[1] //N (to get a numerical value)
Out[1] = 1/2 + I*14.134...
Trying this in Sage, I get:
sage: lcalc.zeros(1)
[14.1347251]
Why does Sage not do the sensible thing like Mathematica and return
the complex number 0.5 + I 14.1347251 ? It would seem much more
logical.
Of course, it is not proven that the real part is 1/2, so how would
the case be handled if a root was not found to have a real part of
1/2 ?
I believe both algorithms assume the Riemann hypothesis in computing
them (otherwise, for example, it would be ambiguous to talk about the
n-th zero anyways). I would guess the reason that lcalc returns the
imaginary part only is that otherwise the first thing one would do to
actually do anything interesting with this data would be to take the
imaginary part, so this just saves the effort and overhead.
- Robert
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