Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?
[Math] Chebyshev’s bias-conjecture and the Riemann Hypothesis
analytic-number-theorynt.number-theoryprime numbersriemann-hypothesis
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Best Answer
Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$: $$ \lim_{x\to\infty} \sum_{p\ge3} (-1)^{(p-1)/2} e^{-p/x} = -\infty. $$ It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function $$ L(s,\chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + \cdots $$ corresponding to the nonprincipal character (mod 4).