Write the divisors of n as 1=d_1 1, put h_α(n)=∑_i<τ(n)(d_i+1/d_i-1)^α. Is liminf_n→∞h_α(n)<∞?

Erdős Problem #1099 — Bounded power sums of consecutive divisor ratios

About 45 years old · traced to 1981

Write the divisors of $n$ as $1=d_1<⋯<d_{τ(n)}=n$ . For $α>1$ , put $h_α(n)=\sum_{i<τ(n)}(d_{i+1}/d_i-1)^α$ . Is $\liminf_{n→∞}h_α(n)<∞$ ?

References

Primary source

P. Erdős, Some problems and results on additive and multiplicative number theory, in Analytic Number Theory (Philadelphia, 1980), Lecture Notes in Mathematics 899 (1981), 171–182.

Additional references

P. Erdős, Some problems and results on additive and multiplicative number theory, in Analytic Number Theory (Philadelphia, 1980), Lecture Notes in Mathematics 899 (1981), 171–182.

divisors number-theory

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