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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)?

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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).

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    $\begingroup$ What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219. $\endgroup$ – KConrad Jan 5 at 4:17
  • $\begingroup$ Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad $\endgroup$ – Greg Martin Jan 5 at 8:48
  • $\begingroup$ @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $\exp\{-2\pi i u\}$? $\endgroup$ – kodlu Jan 6 at 1:35
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    $\begingroup$ @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} \rightarrow 0$ as $p \rightarrow \infty$ through the primes; the series converges for each $x$. Using something like $e^{-2\pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge. $\endgroup$ – KConrad Jan 6 at 16:47
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    $\begingroup$ Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $\sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} \rightarrow \infty$ as $c \rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/… $\endgroup$ – KConrad Jan 6 at 16:51
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Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:

[..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${\Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${\Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.

See also Rubinstein and Sarnak MR review here.

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