On Dirichlet's divisor problem | Zendy

J. R. Wilton | Zendy

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On Dirichlet's divisor problem

Author(s) -

J. R. Wilton

Publication year - 1931

Publication title -

proceedings of the royal society of london series a containing papers of a mathematical and physical character

Language(s) - English

Resource type - Journals

eISSN - 2053-9150

pISSN - 0950-1207

DOI - 10.1098/rspa.1931.0190

Subject(s) - mathematics , integer (computer science) , combinatorics , divisor (algebraic geometry) , prime factor , dirichlet distribution , prime (order theory) , euler's formula , greatest common divisor , mathematical analysis , computer science , boundary value problem , programming language

1. Letd (n ) denote the number of divisors of the positive integern , so that, ifn =p 1 a 1 . . .pr ar is the canonical expression ofn in prime factors,d (n ) = (1 +a 1 ) . . . (1 +ar ), and letd (x ) = 0 ifx is not an iteger; then if (1. 1) D (x ) = Σ'n ≤x d (n ) = Σn ≤x d (n ) ─ ½d (x ), and (1. 2) Δ (x ) = D (x ) ─x logx ─ (2C ─ 1)x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ (x ) = O (√x ),

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