That's not what periodic means. And, no, the zeta function is not periodic (in the usual sense).

@LoïcTeyssier I mean periodic with respect to $\beta$. If the remiann hypothesis is true then the roots satisfy this? Why not other vaues? It would seem weird if only just the roots did

@LoïcTeyssier $\beta\neq \beta^\prime$

@LoïcTeyssier periodic functions of complex variables are not really defined.

True, it is not "periodic" in the usual meaning of the word. But what about the question asked: Is it true that for every $(\alpha, \beta)\in (0,1)\times \mathbb{R}$ there exist $\beta^\prime\neq \beta$ such that $\zeta(\alpha + \beta i)= \zeta(\alpha + \beta^\prime i)$?

@GeraldEdgar I dont think question is asked incorrectly? It means that if you fix $\alpha$ you can find many values of $\beta^\prime \neq \beta $ such that $\zeta(\alpha +\beta i) = \zeta(\alpha + \beta^\prime i)$. You need to think of $\alpha$ as a fixed quantity.

@GeraldEdgar if this is true then $\beta$ seem to have little affect on the value the series actually converges to. It's mostly controlled by $\alpha$. Maybe there are vertical lines for lot of values of $\zeta$ not just the roots

The question is OK, as long as you remove the word "periodic".

@GeraldEdgar it is clear what I mean. Periodic seem to be the right word here

@free_lancer: No. What you are asking is if $\zeta(s)$ is (non-)injective on imaginary (vertical) lines. Moreover, there is an honest notion of periodicity in complex analysis, e.g. en.wikipedia.org/wiki/Elliptic_function .

$\zeta$ is almost periodic like any Dirichlet series (in the critical strip it is a bit trickier but one can use approximations by Dirichlet polynomials or $\eta$ and the periodicity of the factor); $\zeta$ obviously cannot be periodic on any vertical line (for example in the critical strip, it is known that $|\zeta(a+it)| \to \infty, t \to \infty$ and even more is known - eg Selberg limit theorem and the analogues which give precise (probabilistic of course) results about the distribution of $\zeta$ on vertical lines; Titchmarsh is a good starting point

I would expect the answer to be no. Heuristically, for any given $\alpha$, the set of pairs $(\beta,\beta')$ such that $\zeta(\alpha +\beta i) = \zeta(\alpha + \beta^\prime i)$ should be discrete, hence countable. I'm not sure how one would be able to prove it though.

@Conrad its strange if its only repeating at its roots. The value $\beta$ hardly effect the convergence of the series it just effects how much the partial sums ossiculate around the point of convergence. If you fix $\alpha$ and examine the partial sums of its imaginary or real part for many points for $\beta$ it's like a wave meaning $s_n\le Re(\zeta(s))\le s_{n+1}$ or $s_{n+1}\le Re(\zeta(s))\le s_{n}$ with $s_n$ the nth partial sum of each real part.

@Wojowu it is known (and not hard to prove) since Bohr that for $1/2 < a <a_1 <1$ any non zero complex value $w$ is taken at least $c(a,a_1, w)T$ times by $\zeta$ in the corresponding rectangle ($[a, a_1]x[0, T]$) up to height $T$ for $T$ large enough universal - see Theorem 11.10 in Titchmarsh

note that for $n >>\beta$ one has the approximation of $\zeta(\alpha+i\beta)$ by the corresponding Dirichlet polynomial with oscillating term $\frac{n^{1-\alpha-i\beta}}{1-\alpha -i\beta}$ and error $n^{-\alpha}$ so actually the partial sums oscillate with infinite amplitude for fior fixed $\beta$; for $\eta$ same result holds but without the oscillating terms}

@Wojowu the point is that indeed any nonzero values repeat infinitely many times (with known growth rate of repetitions) in any fixed (however small) strip strictly included in the critical strip; in particular it is not inconceivable that some values could repeat on the same line; if for example one could prove that a potential zero would repeat with same frequency, one would prove RH by the well known density theorems

@Conrad how can it ossiculate with infinite amplitude and converge. The behavior is similar to an alternating series with decreasing amplitudes. I plot the partial sums. It like a wave which changes. The imaginary part and real part involve cosine and sine functions and logarithm. The amplitude of each wave is decreasing every time the $cos(log)$ and $sin(log)$ changes. You fix $\alpha$ and $\beta$ does nothing besides changes how much it oscillates around the point it converges to.

@Conrad I'm not denying that values repeat. I'm only saying that it seems to me like only countably many of them can repeat. It is definitely not the case that all complex values occur on every line, which is something your heuristic may suggest.

@Wojowu - the values on any line (between $1/2$ and $1$ strictly) are dense in the complex plane as again is well known so who knows about strict repetition, but clearly for fixed $1/2<a<1$ , any complex number $w$ and any $\epsilon$ one can find $t_n \to \infty$ st $|\zeta(a+it_n)-w| < \epsilon$ so again it is not inconceivable that some such $w$ can repeat infinitely many times- but the result that all nonzero values repeat as noted in any fixed however small strip is well known

@free not sure exactly that we are talking about the same things; if we talk about the partial sums of $\eta(s)=\sum (-1)^{n-1}n^{-s}$, it is true that those approximate the function, though again the approximations start (in general) only from $n >> \beta$ as is not hard to prove using mean value theorems; if we talk about the partials of $\zeta$ itself, those oscillate to infinity as noted

@Conrad they seem these points are logarithmic related. Well at least my intuition would suggest they are far apart because the log is slow. So for each $\beta$ the next $\beta^\prime$ with $\beta^\prime>\beta$ is far away from $\beta$ itself.

that is highly not true on the critical line as zeroes bunch together (for large $\beta$ the next zero is about $1/\log \beta \to 0$ away; also by the theorems mentioned above, for any $1/2<a<a_1<1$ and any $w \ne 0$ the values of $a \le \sigma_k \le a_1, 0 \le \beta_k \le T$ for which $\zeta(\sigma_k+i\beta_k)=\zeta(\sigma_j+i\beta_j)=w, k =1,..\approx T$ are $1$ apart on average, so again not sure what you mean by far away

@Conrad oops yea I plot the functions $\sum_{n=1}^{m}{\dfrac{(-1)^{n-1}\cos(\beta\ln(n))}{n^{\alpha}}}$ and $\sum_{n=1}^{m}{\dfrac{(-1)^{n-1}\sin(\beta\ln(n))}{n^{\alpha}}}$ for different value $\alpha$ and $\beta$ and $m$.

try and plot those for large $\beta \approx 100000$ say and $n \approx 100$ and see what happens

@Conrad Im a bit backwards sometimes. Then they are getting closer together each time we find another. So if $\beta_1>\beta_2>\beta_3$ with $\zeta(\alpha + \beta_1 i ) = \zeta(\alpha + \beta_2 i ) = \zeta(\alpha + \beta_3 i ) $ then $|\beta_1-\beta_2|< |\beta_2-\beta_3| $

you do not know that $\zeta$ is same, just that approximations look same (for $\eta$ but do not forget that there is a periodic factor that changes $\zeta$ so where $\eta(\alpha+i\beta_1) \sim \eta(\alpha+i\beta_2)$, the actual $\zeta$ may be different - try plug in Wolfram alpha and see what happens

@Conrad it still converge the period of sine and cosine is large yes and yes it just effect how much it bounced around the point it converge to. It ossiclate a lot or a little. But it still wave like. As $m$ increases to a value larger then $\beta$ then it start to dampen again

again not sure what we are talking about - the partial series for $\eta$ or for $\zeta$; the first converge, the latter diverge in the critical strip as is very easy to prove

@Conrad the partial sums i provide above in comments with cosine and sine. These are not the imaginary and real part of $\zeta$ is critical strip?

@Conrad the infinitude not important we know it oscillate and converge. The wave behavior is important because that help identifying this behavior I point out.

@Conrad we can use that fact the function is in-between the partial sums and the mean value theorem or some type of continuity together maybe to show it must cross the point it already crossed before

@Conrad if we fix $\alpha$ then we ask if $\zeta(\alpha + \beta i)$ is continuous with respect to $\beta$. In which case there must exist a $\beta^\prime$ so that $\zeta(\alpha +\beta^\prime i) = \lambda$ for every $\lambda$ that is between the partial sum $s_m$.

The length of this discussion suggests that perhaps the question needs to be rewritten to be more focused, and be less of a "moving target". People are not mind-readers.

@YemonChoi what do you suggest? I think these comments are quite helpful towards understanding a possible route to answer the question provided.

@YemonChoi I don't think a "moving target" is an issue here, as written the question is very clear and concrete. The discussion above seems to be about some loosely related topics, like analogous behaviors for partial sums, which I'm not sure are directly helpful in resolving the question.

"Voronin universality" might be interesting to you.

@free the mean value theorem doesn't hold for complex valued functions - first what does it mean that a complex number is between other two is unclear and then for interpretations like using modulus or being on the segment joining the points, neither hold even for analytic functions

@Conrad sorry I mean that the point of convergence for the real and imaginary part as it is between the partial sums. The value $\Re(\zeta(\alpha+\beta i))$ should satisfy $S_m\le Re(\zeta(\alpha+\beta_i))\le S_{m+1}$ or $S_{m+1}\le Re(\zeta(\alpha+\beta_i))\le S_{m}$ for each $m$ with $S_m$ denoting the partial sums of the real part of the $\zeta(s)$ provided by the formula $S_m = \sum_{n=1}^{m}{\dfrac{(-1)^{n-1}\cos(\beta \ln(n))}{n^{\alpha}}}$.

@paulgarrett possible ill look into it

@Conrad if $\Re(\zeta(\alpha + \beta i)))$ is a continuous function of $\beta$ it should cross every value between $S_m$ and $S_{m+1}$.

The question here seems to be related to this earlier question. It's easy to show there's at least one other case other than the zeros of $\zeta(s)$. If $\Im(\zeta(\alpha+i \beta))=0$, then $\zeta (\alpha+i \beta)=\zeta (\alpha-i \beta)$ independent of whether or not $\Re(\zeta(\alpha+i \beta))=0$.

Yes the real part takes any value you want including zero infinitely many times on any vertical line in the critical strip but that is true in very general situations

@Wojowu is seem odd for a function to be periodic only at its roots no?

@Conrad I was also thinking we can analyze the partial derivative with respect to $\beta$ meaning look at $$\dfrac{\partial}{\partial \beta }\Re(\zeta(\alpha+\beta i)) = 0$$ and $$\dfrac{\partial}{\partial \beta }\Im(\zeta(\alpha+\beta i)) = 0.$$ These are critical points of $\zeta(s)$ on the line $\alpha$.

@Wojowu I meant odd for it to be periodic for only one value of $\alpha$

@Conrad let $\lambda$ be a real number and look for solutions to $\lambda = \Re(\zeta(\alpha + \beta~ i))$. For example $\sqrt{\pi} = \Re(\zeta(\alpha + \beta ~i)) $. We can ask which $\alpha$ does this make since for and if they also repeat on a vertical line? It could be possible the solutions just repeat on a vertical line not just the roots. Maybe all solutions to the equation $$ \Re(\zeta(\alpha + \beta~ i))=\lambda = \Im(\zeta(\alpha + \beta ~i))$$ live on a vertical line. Basically, whenever the imaginary and real parts are equal to one another.

@Conrad if we can show multiple vertical lines exist when imaginary and real part are equal then immediately there would be infinitely many points that produce periodic behavior. So it make since to look at $\Re(\zeta)-\Im(\zeta) = 0$

@conrad it possible the solutions to the equation $\Re(\zeta)-\Im(\zeta) = 0$ live on a vertical line or this is just another coincidence for when they both are zero.

@free_lancer You are again misusing the term "periodic" but never mind that. For any $\alpha$ there are doing to be $\beta,\beta'$ with $\zeta(\alpha+\beta i)=\zeta(\alpha+\beta' i)$, it's definitely not just the roots. But this is a different question than "for every $\alpha,\beta$ there is $\beta'$ with ...".

@Wojowu no $\alpha$ is fixed here and $\beta$ is changing. But sorry again its just so natural to say that word

@Wojowu we can ask similar question and look for solutions to the equation $\Re(\zeta(s)) = \Im(\zeta(s)) = \lambda\neq 0 $. Why is it a vertical line only when $\zeta(s)= 0$ (assuming Reimann is correct).

You can ask many different questions, but they are different questions. In an MO post, you should focus on one. Comments to a post are not a place to ramble on about the infinitude of its variants.

@Wojowu sorry i was trying to find route to prove it. I will not comment anymore.

$\Re \zeta =\Im \zeta$ (and neither zero) will happen often since it just means that the argument of $\zeta$ is $\pi/4$ modulo $2\pi$ and the argument is again a real function that is continuous except at zeta zeroes where it jumps by a multiple of $2\pi$ and oscillates a lot from minus infinity to infinity (in the sense that it gets very negative and very positive for arbitrary high abscissas) so it will often pass through $\pi/4+2k\pi$

@Conrad, so the answer to the OP's question is affirmative, by the intermediate value theorem?

they closed this lol