I encountered the following problem in the lecture note in my complex analysis class:
Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$.
Hint. $|(1 − t)|e^t$ is decreasing for $0 < t < 1$ and increasing for $t > 1$. Use this to compare the derivative of a product vanishing at $n \log n$ to one vanishing at $n$, for integers $n$. There is also an easier way using an elementary function that vanishes at the integers.
This is an exercise in the chapter where the Mittag-Leffler theorem and the Weierstrass factorization theorem were introduced. I tried to follow the hint and was able to indirectly construct such a function (by mimicking the proof of Mittag-Leffler), but I was unable to find an explicit closed form.
Would anyone help me find an explicit example? Thanks in advance!