A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

Ramanujan's claim on page 98, in the book (`Ramanujan's note book part 1' by Bruce C. Berndt) is that $$\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,\qquad p>0\tag{1}\label{theclaim}$$ The proposed proofs for $p=1,2$ are reported to be incorrect. For $p>2$ the claim \eqref{theclaim} has been disproved. In the year 1996, my proofs for $p=1,2$ were evaluated to be correct by American Math. Monthly, however, similar proofs were told to have been published in the year 1995, somewhere. The result (1) being asymptotic $x$ needs to be positive and large.

The question here is: What is the latest about this result when $p\in (0,2]$ any information or proof is welcome.

Solution 1:

Proof for $p=1,2$

Euler's constant $\gamma=0.577215...$ is defined as [1] \begin{equation} \sum_{k=1}^{n} \frac{1}{k}\sim\ln n+\gamma ~~~~(1) \end{equation} Ramanujan claimed [2] that \begin{equation} \sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0.~~~~(2) \end{equation} For $p=1,2$ the propose proofs were found incorrect. For $p>2$, the result (2) was disproved. Here the intent is to prove and show that for $p=1,2$ the result (2) is true and consistent with other existing results.

Case 1: Let us quote ([1]: p 955]) \begin{equation} \gamma=-\int_{0}^{\infty} \left( e^{-t}-\frac{1}{1+t} \right) \frac{dt}{t}. \end{equation} We can re-write it as \begin{equation} \gamma=-\lim_{x \to \infty} \left(\int_{1/x}^{x} \frac{e^{-t}-1}{t}+\int_{1/x}^{x} \frac{dt}{1+t}\right). \end{equation} \begin{equation} \gamma=\lim_{x \to \infty}\left( \int_{1/x}^{x} \sum_{k=1}^{\infty} (-1)^{k-1} \frac{t^{k-1}}{k!}dt-\ln x \right), \end{equation} when $x$ is very large, we prove that \begin{equation} \sum_{k=1}^{\infty}(-1)^{k-1} \frac{x^k}{k (k!)} \sim \ln x+\gamma. \end{equation} Case 2: Let us use ([1]: p 955) \begin{equation} \gamma=1-\int_{0}^{\infty} \left( \frac{\sin t}{t}-\frac{1}{1+t} \right) \frac{dt}{t}. \end{equation} \begin{equation} \gamma=1-\lim_{x \to \infty} \int_{1/x}^{x} \left( \frac{\sin t}{t}-\frac{1}{1+t} \right) \frac{dt}{t} \end{equation} \begin{equation} \gamma= \lim_{x \to \infty} \left( 1+\ln x-\int_{1/x}^{x} \frac{\sin t}{t^2} dt\right) \end{equation} Introduce $\sin x=2\sum_{n=0}^{\infty} (-1)^k J_{2k+1}(x)$ ([1]: p 988]) and write \begin{eqnarray} \gamma= \lim_{x \to \infty} \left( 1+\ln x-2\int_{1/x}^{x} \frac{J_1 (t)}{t^2} dt\right)\\ \nonumber-2\sum_{k=1}^{\infty} \int_{0}^{\infty} (-1)^k \frac{J_{2k+1}(t)}{t^2}dt. \end{eqnarray} Further by using ([1]: p 707), when $-\nu-1 <\mu <1/2$. \begin{equation} \int_{0}^{\infty} x^\mu J_{\nu}(x) dx=2^{\mu} \frac{\Gamma(1/2+\nu/2+\mu/2)}{\Gamma(1/2+\nu/2-\mu/2)}~~~~(*). \end{equation} we obtain \begin{eqnarray} \gamma{=}\lim_{x \to \infty} \left( 1+\ln x-2\int_{1/x}^{x} \frac{J_1 (t)}{t^2} dt\right)\\ \nonumber{-}\frac{1}{2} \sum_{k=1}^{\infty} (-1)^k \frac{1}{k(k+1)}. \end{eqnarray} Let us integrate by parts taking $J_1(x)$ as first function and use $2J_1'(x)=J_0(x)-J_1(x)$. The infinite series occurring in above is nothing but $1-2\ln 2$. We get \begin{eqnarray} \gamma{=}\lim_{x \to \infty} \left( 1+\ln x-2\int_{1/x}^{x} \frac{J_1 (t)}{t^2} dt-\int_{1/x}^{x}\frac{J_1(t)}{t}dt\right)\\ \nonumber +\int_{0}^{\infty} \frac{J_1(t)}{t} dt-\frac{1}{2}+\ln 2. \end{eqnarray} Note that $J_1(x) \approx x/2$, for small values of $x$ and for large values $x \sim \infty, J_1(x)=\sqrt{\frac{2}{\pi x}} \cos (x-3\pi/4) \rightarrow 0$. Using (*) again, we get \begin{equation} \gamma=\lim_{x \to \infty} \left( \ln 2x -\int_{1/x}^{x} \frac{J_0(t)}{t} dt\right). \end{equation} Finally, we use the series $J_0(x)=\sum_{0}^{\infty} (-1)^k \frac{(x/2)^{2k}}{(k!)^2}$, to write \begin{equation} \gamma= \lim_{x \to \infty} \left ( \ln 2x -2 \ln x+\frac{1}{2} \sum_{k=1}^{\infty} (-1)^{k-1} \frac{(x/2)^{2k}}{k(k!)^2} \right). \end{equation} Lastly, we get \begin{equation} \sum_{k=1}^{\infty} (-1)^{k-1} \frac{(x/2)^{2k}}{k (k!)^2} \sim 2 \ln (x/2)+2 \gamma, \end{equation} which is nothing but (2), for $p=2$.

$[1]$: I.S. Gradshteyn and I. M. Rhyzik, ``Table of Integrals", Series and Products (Academic Press(NY),1994).

$[2]$: Bruce C. Berndt, ``Ramanujan's Notebooks" Part 1, (Springer, Berlin, 1985) pp 98-100.