Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $a_n$ be a sequence of non-negative numbers. Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$

Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p}{X/\log X}\leq 1?$$

share|improve this question
3  
Just use partial summation to remove the log. –  Lucia Jun 13 at 16:00

1 Answer 1

Here is a direct argument avoiding partial summation. Let $Y:=X/\log^2 X$, then $$ \sum_{p\leq X}a_p = \sum_{p\leq Y}a_p + \sum_{Y<p\leq X}a_p\leq \sum_{p\leq Y}a_p\frac{\log p}{\log 2} + \sum_{Y<p\leq X}a_p\frac{\log p}{\log Y} $$ $$ \leq (1+o(1))\left(\frac{Y}{\log 2}+\frac{X}{\log Y}\right) = (1+o(1))\frac{X}{\log X}.$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.