Chebyshev's Mémoire sur les nombres premiers — §1

§1.[1] All questions which depend on the law of distribution of prime numbers in the series

\[ | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \text {etc. } \]

present in general great difficulties. What we manage to conclude with a very high probability from the tables of prime numbers remains most often without rigorous proof. — For example, the tables of prime numbers lead us to believe that for \( a > 3, \) there is always a prime number greater than \( a \) and less than \( 2a-2 \) (which is known as the postulate of M. Bertrand *); but so far the proof of this proposition has failed for values ​​of \( a \) which exceed the limit of our tables. The difficulty becomes even greater when we give ourselves narrower limits, or when we ask to assign the limit of \( a \) above which the series

\[ a + 1, a + 2,......2a - 2 \]

contains at least two, three, four, etc. prime numbers.

There is another category of very difficult questions which also depend on the law of distribution of the prime numbers in the series.

\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, \text{ etc. } \]

and whose resolution is very necessary. These are the questions on the numerical value of the series whose terms are functions of the prime numbers

\[ 2, 3, 5, 7, 11, 13, 17, \text{ etc. } \]

Euler proved that the series

\[{1 \over 2^a} + {1 \over 3^a} + {1 \over 5^a} + {1 \over 7^a} + {1 \over {11}^a} + {1 \over {13}^a} + \text{ etc. } \]

diverges for the same values ​​of \( a \) for which the series

\[{1 \over 2^a} + {1 \over 3^a} + {1 \over 4^a} + {1 \over 5^a} + {1 \over 6^a} + {1 \over 7^a} + \text{ etc. } \]

diverges; namely, for \( a \leq 1. \) But for certain forms of the term \( u_n, \) the convergence of the series

\[ u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + \text{ etc. } \]

is no longer a necessary condition for the series

\[ u_2 + u_3 + u_5 + u_7 + u_{11} + u_{13} + \text{ etc. } \]

to have a finite value. Such is the case, for example, when \( u_n = {1 \over {n \log{n}}}. \) Indeed, the value of the series

\[ {1 \over {2 \log{2}}} + {1 \over {3 \log{3}}} + {1 \over {5 \log{5}}} + {1 \over {7 \log{7}}} + \text{ etc., } \]

does not surpass \( 1.73, \) as will be proven later, while the series

\[ {1 \over {2 \log{2}}} + {1 \over {3 \log{3}}} + {1 \over {4 \log{4}}} + {1 \over {5 \log{5}}} + {1 \over {6 \log{6}}} + \text{ etc., } \]

is divergent. So what is the criterion for the convergence of series which are composed only of terms with prime numbers \( 2, 3, 5, 7, 11, \) etc.? And, in the case of their convergence, how does one assign the degree of approximation of their values when calculated using only the initial terms? The resolution of these questions in relation to the series of the form

\[ u_2 + u_3 + u_5 + u_7 + u_{11} + u_{13} + \text{ etc. } \]

is very interesting, because they are encountered in some research on numbers.

This memoir contains the resolution of the these questions. I achieve this by treating the function which denotes the sum of the logarithms of the prime numbers below a given limit. From an equation that this function satisfies, we can assign two limits between which the value of this sum falls. Among the various conclusions that we draw from this, we manage to assign limits between which we always find at least one prime number, which leads us very simply to prove Bertrand's postulate. — As for the evaluation of the series of the form

\[ u_2 + u_3 + u_5 + u_7 + u_ {11} + \text{ etc., } \]

we find the criterion to judge whether they are convergent or divergent, and in the first case we give the method to calculate, with a certain degree of approximation, the difference of the value of these series from the sum of their first terms. We also give a formula to calculate, by approximation, how many prime numbers there are that do not exceed a given value, and we assign the limit of the error of this formula, which we could not do until present. In a memoir which I had the honor to present to the Academy of St. Petersburg in 1848, I proved that, if we reject, in the expression of the totality of the prime numbers which do not not surpass \( x, \) all terms that are zero with respect to

\[ {x \over {\log{x}}}, \;\; {x \over {\log^2{x}}}, \;\; {x \over {\log^3{x}}}, \;\; \text{ etc., } \]

when we make \( x = \infty, \) this expression reduces to \( \int_{2}^{x} {dx \over \log{x}} \); but for finite values ​​of \( x \) one is uncertain of the value of the terms rejected. As for Legendre's formula, its degree of approximation is only known within the limits of the tables of prime numbers used to verify it.

*) Journal de l'école polytechnique, cahier XXX.

— end §1—

Translated by Mike Bertrand

November 5, 2020

Translator's note: This is §1 of Chebyshev's Mémoire sur les nombres premiers (Memoir on Prime Numbers) from 1852. §2 — §5 were translated into English by J. D. Tamarkin and appeared in A Source Book in Mathematics, edited by David Eugene Smith and published in 1929. A mere twenty pages long, the memoir established key methods and results, including Bertrand's postulate (or conjecture), and is highly readable to this day. It is broken into nine sections:

§1. An introduction to the whole — this is what is translated into English above (original)

§2. Definition and basic results for \( \theta(x), \psi(x), \) and \( T(x). \) (original, English)

§3. Inequalities involving \( \theta(x) \) and \( T(x). \) (original, English)

§4. Inequalities involving \( T(x) \) and \( \log{x}. \) (original, English)

§5. Inequalities involving \( \psi(x) \) and \( \log{x}. \) (original, English)

§6. Proof of Bertrand's postulate. (original)

§7. Proof that \( \sum_{m=2}^\infty {F(m) \over \log{m}} \) converges if and only if \( \sum_{p \text{ prime }} F(p) \) does, given simple conditions on \( F. \) (original)

§8. Proof that \( \sum_{p \text{ prime }} {1 \over {p \log{p}}} \) converges. (original)

§9. Approximation of \( \pi(x). \) (original)

Bertrand's Postulate. Or Bertrand's conjecture, first proposed by Joseph Bertrand in 1845 in an early paper on group theory, then regarded as a matter of permuting variables like \( x_1 \) and \( x_2 \) in polynomial expressions. Bertrand postulated that for any natural number \( n, \) there is always a prime number \( p \) such that \( n/2 < p < n - 2. \) Or equivalently, as Chebyshev says above, for any natural number \( m, \) there is always a prime number \( p \) such that \( m < p < 2m - 2. \) Bertrand checked the postulate in prime tables for values of \( n \) up to six million:

Pour démontrer cette proposition, j'admettrai comme un fait que, quel que soit un nombre \( n \) supérieur à 6, il existe toujours un nombre premier, au moins, compris entre \( n - 2 \) et \( n / 2.\) Cette proposition est vraie pour tous les nombres inférieurs à six millions, et tout porte à croire qu'elle est générale.

To demonstrate this proposition, I will admit as a fact that, for any number \( n \) greater than 6, there always exists at least one prime number between \( n - 2 \) and \( n / 2. \) This proposition is true for all numbers less than six million, and everything suggests that holds in general.[2]

Paul Erdős reproved Bertrand's postulate in 1932 in his first published article at the age of 19.[3] Erdős relied heavily on \( C_{2n} = {2n \choose n}, \) the binomial coefficient in the middle of row \( 2n \) in Pascal's triangle, which seems like magic until you realize the connection with Chebyshev's \( T(n) = \sum_{k=1}^n \log{k} = \log{n!}: \)

\begin{align}
T(2n)-2T(n) &= \log{(2n)!} - 2 \log{n!}\\
&= \log{{(2n)!} \over {(n!)^2}}\\
&= \log{2n \choose n}.
\end{align}

This is in the spirit of Chebyshev, but so much simpler than the expression he relied on:

\[ T(x) + T\left({x \over 30}\right) - T\left({x \over 2}\right) - T\left({x \over 3}\right) - T\left({x \over 5}\right). \]

^ 1. "Mémoire sur les nombres premiers" (Memoir on Prime Numbers), by P. L. Chebyshev, in Oevres de P. L. Tchebycheff, Tome I, edited by A. Markoff and N. Sonin, St.-Pétersbourg (1899), pp. 49-70.

^ 2. "Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme" (Memoir on the number of values that a function can take when the letters it contains are permuted), by Joseph Bertrand, Journal De L'Ecole Royale Polytechnique, Tome I (1845), pp. 123-140. See p. 129 for the postulate.

^ 3. "Beweis eines Satzes von Tschebyschef" (Proof of a theorem of Chebyshev), by Paul Erdős, Acta Universitatis Szegediensis (Szeged, Hungary), 5 (1932), pp. 194-198.

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