Abstract
Let Pn be the n-th prime number. We prove the following double-inequality. For all integers k?5 we have exp[k(c0?loglogk)]? k2 k/p1?p2?...?pk ? exp[k(c1?loglogk)] with the best possible constants c0 = 1/5 log23 + loglog5=1:10298? and c1 = 1/192log(36864/192)+loglog 192?1/192log(p1?p2???p192)=2.04287... This reffines a result published by Gupta and Khare in 1977.
Highlights
The work on this note has been inspired by a remarkable short paper published by Gupta and Khare [4] in 1977
The authors presented a connection between the binomial coefficient k2 k and the product of the first k prime numbers
We demonstrate that (1.2) is valid with c0 = 1.10298 . . . and c1 = 2.04287 . . . . This provides a positive lower bound for Qk, but improves (1.1) for all k ≥ 2237
Summary
The work on this note has been inspired by a remarkable short paper published by Gupta and Khare [4] in 1977. This refines a result published by Gupta and Khare in 1977. The authors presented a connection between the binomial coefficient k2 k and the product of the first k prime numbers. The Proposition implies an interesting number theoretical theorem: if 1794 ≤ k ≤
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