# Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?

The Cramér probabilistic model of primes is built on the assumption that the probability of $$n$$ being prime is

$$\Pr(n)=\frac{1}{\log (n)}$$

This is not a big leap from the Prime Number Theorem which tells us that the density of primes is approximately $$1/\log(x)$$, an approximation that is asymptotically more accurate for larger $$x$$.

My question is, was Harald Cramér the first to use this density as a probability, in the 1930s?

His famous paper was published 1936/7 (pdf).

• Gauss had discovered this much earlier (the density of primes near a large $n$ is around $1/\log n$) and it was the reason he conjectured PNT with the approximation $\int_2^x dt/\log t$. Cramer was the first to apply nontrivial theorems in probability theory to conjecture properties of the primes using that density model. – KCd May 4 at 13:57

A good place to look is Granville’s paper “Harald Cramér and the distribution of primes numbers.” It is on Granville’s website here. He brings in Cramér’s work starting on the bottom of page 19. The reason Cramér deserves to have this probabilistic model for the primes named after him is that he went much further than anyone before him in exploring consequences of the basic $$1/\log n$$ heuristic to primes. He applied nontrivial theorems from probability theory to make predictions about the primes in a style nobody had ever done before. This did not directly prove anything about the primes, and Cramér acknowledged this, but simply having an idea of what might be true is often very important in trying to understand how mathematical patterns fit together.

As Professor K. Conrad mentions, the short answer to this question is NO.

Around 1792 Gauß already knew that the "frequency [of the primes] is on the average inversely proportional to the logarithm, so that the number of primes below a given bound $$n$$ is approximately equal to $$\int \frac{1}{\log n} \, dn,$$ where the given logarithm is understood to be hyperbolic".

Gauß mentioned this investigations of his to Johann Franz Encke in a letter dated December 24, 1849. The said letter can be found here:

https://cims.nyu.edu/~tschinke/princeton/.gauss/Briefe-B.pdf

If I recall correctly, a complete translation into English of this letter appeared at the end of the following article:

L. J. Goldstein, A history of the prime number theorem, Amer. Math. Monthly 80 (1973), 599–615. MR 313171, https://doi.org/10.2307/2319162

If you don't have access to the above article, you may want to take a look at this 2005 note of Y. Tschinkel in BAMS.

• Thanks. Cramér's model is named after him, and I wonder why that is, because that model is the very simple model that assumes $n$ is prime with probability $1/\ln(n)$. Have I misunderstood what Cramér's model is? Why is this probabilistic model not called Gauss' model? – Tariq May 4 at 21:18
• Quite possibly the model was named that way because of the fact that "Cramér was the first to apply nontrivial theorems in probability theory to conjecture properties of the primes using that density model" (I am quoting Professor Keith Conrad here)... The issue about the name might be simply attibuted to Stigler's law of eponimy, too: en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy – José Hdz. Stgo. May 4 at 22:02
• Thanks. Where can I read more by Prof Conrad, who seems an authority on this? – Tariq May 4 at 22:43