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convexity conjecture
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(Conjecture)
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(Hardy & Littlewood) Given integers
 , it is never the case that
 , where  is the prime counting function.
For example: There are 269 primes below 1729. There are 304840 primes below 4330747. If we add up these values of the prime counting function, we get 305109. This is more than
 .
Crandall and Pomerance believe this conjecture to be false but also that any counterexample is way too large to be discovered today. If we limit ourselves to 100 for both variables,
 tends to fall in the range
 .
- 1
- R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001: 1.2.4
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"convexity conjecture" is owned by PrimeFan.
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(view preamble)
| Other names: |
Hardy-Littlewood convexity conjecture |
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Cross-references: range, variables, limit, counterexample, conjecture, primes, prime counting function, integers
There is 1 reference to this entry.
This is version 2 of convexity conjecture, born on 2007-02-25, modified 2007-02-28.
Object id is 8991, canonical name is ConvexityConjecture.
Accessed 782 times total.
Classification:
| AMS MSC: | 11A41 (Number theory :: Elementary number theory :: Primes) |
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Pending Errata and Addenda
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