Dirichlet formula

Dirichlet formula for the number of divisors - asymptotic formula

\sum _{n\leq N}\tau (n)=N\ln N+(2\gamma -1)N+O({\sqrt {N}}),

{\ displaystyle \ sum _ {n \ leq N} \ tau (n) = N \ ln N + (2 \ gamma -1) N + O ({\ sqrt {N}}),}

\ sum _ {{n \ leq N}} \ tau (n) = N \ ln N + (2 \ gamma -1) N + O ({\ sqrt N}),

Where $\tau (n)$ ${\ displaystyle \ tau (n)}$ $\ tau (n)$ - number of dividers $n$ ${\ displaystyle n}$ $n$ , $\gamma$ ${\ displaystyle \ gamma}$ $\ gamma$ - Euler constant - Mascheroni , and $O$ ${\ displaystyle O}$ $O$ - O-big .

About proof

The proof immediately follows from the fact that the sum indicated is equal to the number of integer points with integer positive coordinates in the domain bounded by the hyperbola $yx=N$ ${\ displaystyle yx = N}$ and coordinate axes.

History

The formula was obtained by Dirichlet in 1849.

Dirichlet formula

About proof

History

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