Abstract: For any positive integer x,let π（ｘ） denote the number of primes p with ｐ≤ｘ.Further,for any positive integers n,let ｆ（ｎ）＝π（ｘ）＋π（２ｘ）＋…＋π（ｎｘ）.In this paper，it is proved that if ｘ≥４　and ｎ≥６,then ｆ（ｎ）＞π（ｎ（ｎ＋１）ｘ／２）.

Key words: prime；Chebyshev&rsquo, s theorem；inequality