| viXra.org > Number Theory > viXra:2604.0068 |
 |
 |
| viXra.org > Number Theory > viXra:2604.0068 |
|
|
Number Theory |
|
Authors: Rusin Danilo Olegovich
We introduce and study the entire function $lambda(s) = sum_{n=1}^{infty} n^{s+1/2}/n^n$,defined by a Dirichlet-type series with super-exponential coefficients. We prove that$lambda(s)$ converges absolutely for all $s in mathbb{C}$, uniformly on compact sets,and is therefore an entire function of order zero. We establish a closed-form evaluationof the special value $lambda(-1/2) = int_0^1 x^{-x}, dx$, connecting $lambda$ to theclassical Sophomore's Dream identity of Bernoulli. We further prove that $lambda(s)$ isterm-by-term differentiable, with $lambda^{(k)}(s) = sum_{n=1}^{infty} (ln n)^k n^{s+1/2}/n^n$for all $k geq 0$, justified by the Weierstrass $M$-test. Finally, we propose a conjectureconnecting $lambda(s)$ to the Riemann zeta function.
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Download: PDF
[v1] 2026-04-18 23:35:24
Unique-IP document downloads: 71 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.
| Third-Party Links: |
|