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The motivation for this question is the same as in my previous question in MO: https://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over

Let us consider an analytic function $f$ defined in the whole complex plane which has infinitely many zeros. Let us restrict the function to the interval $(0,1)$ as follow: $g(t)=f(1-2t)$. I look for the number of roots of $g$ in $(0,1)$.

My question is: What I can say for the case of $g$ defined by using $f$ in $(0,1)$.

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    What do you mean by "for analytic function, the number is infinite"? There are analytic functions, such as the exponential function, that have no zeros at all.2012-12-15
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    Only for the function f described here.2012-12-15
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    But you haven't described one? Your edit has very slightly improved things, because it wasn't clear before whether you meant $f$ or $g$, but you still haven't said anything about $f$ that would suggest that it has infinitely many zeros.2012-12-15
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    The link to the MO question, though certainly helpful, hasn't shed any light on my questions above.2012-12-15
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    I see. The formulation in the question was an unusual and misleading way of expressing that; I've edited it to reflect what I understand you meant; please check if I got it right. (By the way, you misspelled my user name; I only happened to get notified because no-one else has commented yet.)2012-12-15
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    joriki: f has infinite number of zeros and I looking for the number for g2012-12-15
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    @Peter: Strange, the edit history doesn't show that edit or its reversion.2012-12-15
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    @user53124: Cross-posting between MO and math.SE is usually not a good idea. If you do cross-post, you should at least wait for answers on one for a while before posting on the other. Most importantly, however, it's very bad style to cross-post without saying so; this leads to entirely unnecessary duplication of efforts. In the present case this is particularly well exemplified by quid's comment in the MO thread, which is almost verbatim identical to mine in this one.2012-12-15
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    @joriki: I am not aware that the two websites are the same. Sorry for this problem.2012-12-15
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    @user53124: They are not the same; that's the problem. You posted on two different websites without telling people on either of them about it, so you made two different groups of people work independently without a chance to profit from each others' progress in answering the question or, in this case, in getting you to clarify it. That's pure waste.2012-12-15

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The number of zeros of $g$ in $(0,1)$ is equal to the number of zeros of $f$ in $(-1,1)$. If you don't know where the infinitely many zeros of $f$ are, you don't know anything about the zeros of $g$.