Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the smallest integer $n$ for which $\theta(n) > n$? Here $\theta(x) = \sum_{p \leq x} \log p$.

I googled around, checked some likely textbooks, and ran a program for $n \leq 10^7$, but didn't find it.

The integer and a reference would be nice.


share|cite|improve this question
Well, assuming there is such a thing. The first number for which $\mbox{Li} x$ and $\pi(x)$ change places is almost $10^{400}$ – Will Jagy Jan 25 '13 at 2:16
@RClark Wikipedia has some good information on the number Will refers to: – Matthew Conroy Jan 25 '13 at 3:02
@MatthewConroy: Thanks, I found that. – RClark Jan 25 '13 at 18:13
@WillJagy Such a number does exist by Littlewood's Oscillation Theorem for $\theta(x)$. – RClark Jan 25 '13 at 18:14
This function is most commonly written $\vartheta$ with \vartheta. – lhf May 6 '13 at 17:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.