Does it exist a given explicit function $f:\mathbb N\to\mathbb R$ such that forall N big enough $\exists m,n\in\mathbb N: m,n>N \wedge f(m)\leq p_m\wedge f(n)\geq p_n$?


A literal answer to your question: $f(x)=p_{\lfloor x\rfloor}$ works. But I think you want something stronger than merely being "explicit", more like a finite-length formula over well-known functions.

Littlewood proved in 1914 that for all $N$ there are $n,m>N$ such that $$ \operatorname{li}(m)\le\pi(m)\ \vee\ \operatorname{li}(n)\ge\pi(n). $$ The usual citation is [1] though I have not read the paper (it is in French). It is likely that for all $N$ there are $n,m>N$ such that $$ ¿\qquad\operatorname{li}^{-1}(p_m)\le m\ \vee\ \operatorname{li}^{-1}(p_n)\ge p\qquad? $$ but I cannot immediately prove this.

[1] J. E. Littlewood, Sur la distribution des nombres premiers, Comptes Rendus 158 (1914), pp. 1869-1872.

  • $\begingroup$ It may be that I don't understand your notation well. I think Littlewood's paper proves something stronger--the inequalities are strict and the theorem guarantees a difference that depends explicitly on x (your m,n). Sidestepping the French, the claim is Littlewood's eq. (3). I realize this is not the main point of your answer... $\endgroup$ – daniel Aug 19 '16 at 7:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.