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Smooth numbers; Riemann zeta function.

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of $\zeta(1/2+it)$ is bounded by $(\log t)^{\frac{1}{2}+o(1)}$, we show that for any given $\alpha > 0$ the interval $(X, X+\sqrt{X}(\log X)^{1/2+o(1)}]$ contains an integer having no prime factor exceeding $X^\alpha$ for all 𝑋 sufficiently large.

Goutam Pal¹ Satadal Ganguly²

1. 5/1, Nandaram Sen, First Lane, Kolkata 700 005, India
2. School of Mathematics, Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Mumbai 400 005, India

Published

November 2010

Manuscript received

15 June 2010

Manuscript revised

13 July 2010