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Properties of the Infinite Sum of a Real Number's Prime Powers

Consider the series $\sum_{i=1}^\infty x^{p_i}$ , where is an arbitrary real number and p_i is the th prime. That is, the series $x^2 + x^3 + x^5 + x^7 + \cdots$ . Equivalently, this series can be written as $\sum_{n=1}^\infty \beta_n x^n$ , where $\beta_n = \begin{cases} 0 & \text{if n not prime} \\ 1 & \text{if n prime} \\ \end{cases}$ . For what values of does this series converge?

Clearly, $|\beta_n x^n| \le |x^n|$ , so by the direct comparison test the series must converge absolutely wherever the geometric series $\sum_{n=1}^\infty x^n$ converges absolutely. That is, the series converges absolutely $|x| \lt 1$ . If $|x| \gt 1$ , then $\lim_{n \to \infty} \beta_n x^n \ne 0$ , so the series diverges. If x=1 , then the series becomes $1+1+1+\cdots$ , diverging to $+\infty$ . if x=-1 , then the series becomes $1-1-1-\cdots$ , diverging to $-\infty$ . We conclude that $\sum_{i=1}^\infty x^{p_i}$ , converges on (-1, 1) and nowhere else on the real line.

We can now define a function $\lambda: (-1, 1) \to \mathbb{R}$ as $\lambda(x)= \sum_{i=1}^\infty x^{p_i}$ . Here is the graph of $y = \lambda(x)$ , with $\lambda(x)$ approximated up to the $x^{41}$ term:

Based on this graph, it would appear that $\lambda$ is continuous, that $\lim_{x->1^-} \lambda(x)$ is $+\infty$ , and that $\lim_{x->-1^+} \lambda(x)$ is $-\infty$ . It also appears that the function is increasing for -values from the left edge of the domain to about -0.5, then decreasing up to about zero, then increasing again up to the right edge of the domain. Let us see how many of these properties we can prove.

Since $\lambda$ is defined by a convergent power series, it is an analytic function, which implies, among other things, that it is continuous. The derivative of $\lambda$ is $\frac{d}{dx} \lambda(x) = \sum_{i=1}^\infty p_i x^{p_i - 1}$ , or $\lambda^{\prime}(x) = 2x + 3x^2 + 5x^4 + 7x^6 + \cdots$ . Since all primes besides 2 are odd, every term in this series except the first is of even order. For $x \lt 0$ , all the even-order terms in the definition of $\lambda^\prime$ are positive and the single odd-order term is negative. Whether $\lambda$ is decreasing or increasing, therefore, depends on whether 2x > 3x^2 + 5x^4 + 7x^6 . Approximately, it seems that $\lambda$ is increasing on (-1, -0.457) and decreasing on (-0.457, 0). For $x \gt 0$ , each term is positive, so $\lambda^\prime$ is positive, so $\lambda$ is increasing. At the moment I am unable to find proofs of the one-sided limits at 1 and -1.