Refinements of Goldbach’s Conjecture, and the Generalized Riemann Hypothesis
نویسندگان
چکیده
Abstract: We present three remarks on Goldbach’s problem. First we suggest a refinement of Hardy and Littlewood’s conjecture for the number of representations of 2n as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach’s conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach’s conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of k primes.
منابع مشابه
Generalized Riemann Hypothesis
(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA])...
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