Very elementary interpretations of the Euler-Mascheroni constant from counting divisors in intervals

R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the number of divisors of q between sqrt{F(q)} and F(q). Then, on the average, Z_F(q) equals Euler's constant Theorem 2 Fix a in (0,1). Write A for the average number of divisors of n that lie in (0,sqrt{a n}) minus the number of that lie in (sqrt{a n},a n)$. Then A= (sum_{i=1}^{\lceil {1-a}/a \rceil} \frac{1}{i}) - ln(1/a).
Comment: 14 pages, 9 figures -->
Druh dokumentu: Working Paper
Přístupová URL adresa: http://arxiv.org/abs/0810.1354
Přírůstkové číslo: edsarx.0810.1354
Autor: Feldman, David V.
Rok vydání: 2008
Předmět:
Druh dokumentu: Working Paper
Popis: Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the number of divisors of q between sqrt{F(q)} and F(q). Then, on the average, Z_F(q) equals Euler's constant Theorem 2 Fix a in (0,1). Write A for the average number of divisors of n that lie in (0,sqrt{a n}) minus the number of that lie in (sqrt{a n},a n)$. Then A= (sum_{i=1}^{\lceil {1-a}/a \rceil} \frac{1}{i}) - ln(1/a).
Comment: 14 pages, 9 figures
Databáze: arXiv
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