1
$\begingroup$

A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. Can someone refresh my memory and provide the link / reference to this statement?

I worked on this topic a while back, and came up with the following formula, valid for a large class of even functions:

$$f(t) = \frac{\sin\pi t}{\pi}\cdot \Big[\frac{f(0)}{t} +\phi'(t)\sum_{k=1}^\infty (-1)^k \frac{f(k)}{\phi(t)-\phi(k)} \Big]$$ where (say) $\phi(t)=t^2$. I posted it on MO a while back, and even proved the formula. I remember the proof was quite simple, but I can't remember any of it. There's quite a bit of details in my earlier question from years ago, on MO, here. It was a question about interpolation.

I am again interested in this topic, and I applied the above formula to the real part of the Riemann zeta function $\zeta(t+i\sigma)$, and it works. It works at least for $0<t<20$ and $\sigma=0.8$. This surprises me a bit since that function does not satisfy the conditions required to guarantee convergence to the right solution (conditions that I found myself but they could be more strict than actually needed).

If you want to answer a question a bit more challenging than my initial question, you can answer whether my formula applies or not to the real part of the Riemann zeta function, even for large values of $t$ and different $\sigma$'s. But at this point I am only interested in a reference to the theorem in question.

Update

I wrote an article unrelated to this topic (it's about fuzzy spatial regression as a machine learning technique), and not yet completed. However Exercise 2 in section 5 deals with the problem described here. You can download the article here. Figure: Dirichlet eta function (real part, bottom) and interpolation error (top)

Improve this question
$\endgroup$
15
  • 4
    $\begingroup$ Possibly the vast topic of the distribution of zeries of entire functions of exponential type contains what you are looking for. A standard monograph is by B. Ja. Levin. $\endgroup$
    – terceira
    May 21, 2022 at 6:56
  • 2
    $\begingroup$ @Steven that's a very loose statement as there are tons of entire functions (or if you prefer meromorphic with a pole at $1$) that have the same values as RZ on any sequence of numbers without an accumulation point; one has to qualify the statement by some other specific properties of RZ $\endgroup$
    – Conrad
    May 21, 2022 at 16:32
  • 2
    $\begingroup$ @Conrad My interest was motivated by the discovery that the coefficients $a(n)$ of $f(x)=\sum\limits_{n=1}^x a(n)$ and $F(s)=s \int\limits_0^\infty f(x)\,x^{-s-1}\,dx=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$ can be determined from the values of $F(s)$ at the positive even integers when $F(s)$ converges for $\Re(s)\ge 2$. See the answer I posted at math.stackexchange.com/q/4160465 to a question on Math StackExchange. $\endgroup$
    – Steven Clark
    May 21, 2022 at 16:59
  • 1
    $\begingroup$ Why do you think that paper is correct? Among other things it claims a proof of RH $\endgroup$
    – Conrad
    May 21, 2022 at 21:02
  • 1
    $\begingroup$ In these topics handwaving easily leads to mistakes so I would first have a riguros proof before claiming any result $\endgroup$
    – Conrad
    May 21, 2022 at 21:15

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.