About the cover: On the distribution of primes—Gauss’ tables

Abstract

In 1791, Carl Wilhelm Ferdinand, the Duke of Brunswick, gave fourteen-yearold Gauss a collection of mathematics books which included logarithm tables by Schulze. These were the first logarithm tables that Gauss possessed. The tables listed decimal logarithms up to 7 digits as well as natural logarithms of all natural numbers up to 2200 and prime numbers up to 10,009 [3]. Gauss worked on extending the tables: a computation of ln(10037) can be found in his papers from that time. It is conceivable that seeing and using the tables of logarithms of primes, Gauss was led to discover the law governing the distribution of primes, the Prime Number Theorem. Throughout his life Gauss returned again and again to this issue, matching data from published tables of prime numbers with his prediction. Here are the first lines of a four-page letter from Gauss to his student Johann Franz Encke, lieutenant of artillery and later astronomer in Berlin, dated December 24, 1849: My distinguished friend, Your remarks concerning the frequency of primes were of interest to me in more ways than one. You have reminded me of my own endeavors in this field which began in the very distant past, in 1792 or 1793, after I had acquired the Lambert supplements to the logarithmic tables. Even before I had begun my more detailed investigations into higher arithmetic, one of my first projects was to turn my attention to the decreasing frequency of primes, to which end I counted the primes in several chiliads and recorded the results on the attached white pages. I soon recognized that behind all of its fluctuations, this frequency is on the average inversely proportional to the logarithm, so that the number of primes below a given bound n is approximately equal to ∫ dn