Mangala J Narlikar - On a theorem of Erdos and Szemeredi

hrj:90 - Hardy-Ramanujan Journal, January 1, 1980, Volume 3 - 1980 - https://doi.org/10.46298/hrj.1980.90
On a theorem of Erdos and SzemerediArticle

Authors: Mangala J Narlikar 1

K. F. Roth proved in 1957 that if 1=q1<q2<q3 is the sequence of square-free integers, then qn+1qn=O(n313(logn)413). P. Erdös set the problem in more general conditions. If 2b1<b2<b3 is a sequence of mutually coprime integers, and 1bi<, he proved that if 1=q1<q2<q3 is the sequence of integers not divisible by any bi, thenqn+1qn=O(qθn),

where 0<θ<1. C. Szemeredi made an important progress and proved that if Q(x)=qix1, then Q(x+h)Q(x)>>h, where hxθ. Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and 1bi<, thenQ(x+h)Q(x)>>h,
where xhxθ and θ>12 is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.


Volume: Volume 3 - 1980
Published on: January 1, 1980
Imported on: March 3, 2015
Keywords: gaps in sequences like square-free integers,[MATH]Mathematics [math]

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