# Thread: Inequality with the prime-counting function

1. ## Inequality with the prime-counting function

Hi there guys, I hope I've posted this in the right section;

I'm going through a paper, and there is one statement which is:

$n^{\pi(n)} < 3^n$

for sufficiently large $n$, where $\pi(n)$ is the prime-counting function defined as the function counting the number of prime numbers less than or equal to $n$. I'm having real difficulty justifying this claim, and I'd be really grateful if someone could help me out.

Thanks in advance,
HTale

2. Originally Posted by HTale
I'm going through a paper, and there is one statement which is:

$n^{\pi(n)} < 3^n$

for sufficiently large $n$, where $\pi(n)$ is the prime-counting function defined as the function counting the number of prime numbers less than or equal to $n$.

Take logs, and it says $\pi(n)<\frac n{\ln n}\ln3$. But ln(3)>1, so the result will follow from the prime number theorem.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8942729234695435, "perplexity": 164.8388228138436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720238.63/warc/CC-MAIN-20161020183840-00197-ip-10-171-6-4.ec2.internal.warc.gz"}