# A curious equality of integrals involving the prime counting function?

This post discusses the integral, $$I(k)=\int_0^k\pi(x)\pi(k-x)dx$$

where $$\pi(x)$$ is the prime-counting function. For example, $$I(13)=\int_0^{13}\pi(x)\pi(13-x)dx = 73$$

Using WolframAlpha, the first 50 values for $$k=1,2,3,\dots$$ are,

$$I(k) = 0, 0, 0, 0, 1, 4, 8, 14, 22, 32, 45, 58, 73, 90, 110, 132, 158, 184, 214, 246, 282, 320, 363, 406, 455, 506, 562, 618, 678, 738, 804, 872, 944, 1018, 1099, 1180, 1269, 1358, 1450, 1544, 1644, 1744, 1852, 1962, 2078, 2196, 2321, 2446, 2581, 2718,\dots$$

While trying to find if the above sequence obeyed a pattern, I noticed a rather unexpected relationship:

Q: For all $$n>0$$, is it true, $$I(6n+4) - 2\,I(6n+5) + I(6n+6) \overset{\color{red}?}= 0$$

Example, for $$n=1,2$$, then $$I(10)-2I(11)+I(12)=32-2*45+58 = 0$$ $$I(16)-2I(17)+I(18)=132-2*158+184= 0$$ and so on.

• Note your proposed equation doesn't hold for $n = 0$ as $I(4) = 0$, $I(5) = 1$ and $I(6) = 4$. – John Omielan Feb 27 '19 at 4:04
• @JohnOmielan: A typo. I meant all $n>0$. I will correct it. – Tito Piezas III Feb 27 '19 at 4:09
• I have checked to confirm what you're asking is true for $n$ up to $18$. However, I have my doubts it'll always work, partially because it doesn't work for $n = 0$. Also, a similar type condition is that $I(6n) - 2I(6n + 1) + I(6n + 2) = 2$, which holds for $1 \le n \le 5$, but at $n = 6$, the LHS becomes $0$ instead. If I get a chance, I will investigate your equation to see if I can figure out why it's true for at least the first $18$ values and, more importantly, will it always stay true. Regardless, though, it's an excellent observation you've made, even if it doesn't always hold. – John Omielan Feb 27 '19 at 4:47
• I checked your result up to $n=533$ (for $n \geq 534$, I have problems. Would you be interested by a huge table of $I(k)$ (I was able to generate it up to $k=540$). This is a very interesting problem. – Claude Leibovici Feb 27 '19 at 5:06
• Up to $k=540$, $I(k)$ is a prime for this list $$\{13,57,119,167,171,173,175,341,395,397,427,431,473,515,519\}$$ – Claude Leibovici Feb 27 '19 at 8:13

The answer is yes. Sketch of solution: $$I(k) = \int_0^k \sum_{p\le x} \sum_{q\le k-x} 1 \,dx = \sum_p \sum_{q\le k-p} \int_p^{k-q} dx = \sum_p \sum_{q\le k-p} (k-(p+q)) = \sum_{m\le k} r(m)(k-m),$$ where $$r(m)$$ is the number of ways of writing $$m$$ as the sum of two primes. Then \begin{align} I(6n+6) &{}-2I(6n+5)+I(6n+4) \\ &= \sum_{m \le 6n+4} r(m)\big((6n+6-m)-2(6n+5-m) +(6m+4-m)\big) + r(6n+5) \\&= 0 + r(6n+5); \end{align} and $$r(6n+5)=0$$ for every $$n\ge1$$, since the only way the odd integer $$6n+5$$ can be the sum of two primes is $$6n+5=2+(6n+3)$$, but $$6n+3=3(2n+1)$$ is always composite when $$n\ge1$$.
The same argument gives $$I(6n+2)-2I(6n+1)+I(6n) = r(6n+1)$$, which is $$2$$ if $$6n-1$$ is prime and $$0$$ otherwise; this is why (as observed by John Omielan) it equals $$2$$ for $$1\le n\le 5$$ but $$0$$ for $$n=6$$.
• Greg, do you know how to address Ultradark's question regarding when $I(k)$ is prime? – Tito Piezas III Feb 27 '19 at 5:31