Some generalizations of Chebyshev's conjecture
Chebyshev asserted in 1853 that (1) \(\lim_{x\to 0+}\sum_{p>2}(- 1)^{(p-1)/2}\) \(e^{-xp}=-\infty\) (p runs over the odd prime numbers) which says that there are more primes \(\equiv 3 (mod 4)\) than \(\equiv 1 (mod 4)\) in the sense of Abel's mean. By classical results of Hardy- Littlewood and Landau we know that (1) is equivalent to the Riemann Hypothesis for the L-function corresponding to the non-principal character \(\chi\) (mod 4). The present author considers relations similar to (1): \[ (2)\quad \lim_{x\to 0+}\sum_{p>2}(-1)^{(p-1)/2}\quad e^{-(xp)^{\alpha}}=-\infty,\quad 0<\alpha <\alpha_ 0,\quad \alpha_ 0\quad fixed. \] \[ (3)\quad \lim_{x\to 0+}\sum_{p>2}(- 1)^{(p-1)/2}\quad K_ 0(2\sqrt{xp})=-\infty \] where \(K_ 0\) denotes the Bessel function of the third kind and index 0. He finds that both (2) and (3) are equivalent to the Riemann Hypothesis for the same L-function (mod 4). Moreover, he states (2) and (3) as conjectures ((3) in a somewhat more general form). The method of proof goes in classical lines and is suitable for some other weighting functions in place of \(e^{- (xp)^{\alpha}}\) and \(K_ 0(2\sqrt{xp})\) in (2) and (3) resp.
scientific article; zbMATH DE number 4083745