In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Poussin proof is the proof of an identity related to the
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...
of a
ratio
In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
.
In 1838,
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...
proved an approximate formula for the average number of divisors of all the numbers from 1 to n:
:
where ''d'' represents the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...
, and γ represents the
Euler-Mascheroni constant.
In 1898,
Charles Jean de la Vallée-Poussin
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning wa ...
proved that if a large number n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is γ:
:
where represents the
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...
of ''x'', and π represents the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal ...
.
For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5.
References
*Dirichlet, G. L.
Sur l'usage des séries infinies dans la théorie des nombres, ''Journal für die reine und angewandte Mathematik'' 18 (1838), pp. 259–274. Cited in MathWorld article "Divisor Function" below.
*de la Vallée Poussin, C.-J. Untitled communication. ''Annales de la Societe Scientifique de Bruxelles'' 22 (1898), pp. 84–90. Cited in MathWorld article "Euler-Mascheroni Constant" below.
External links
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Number theory
{{numtheory-stub