The two functions θ(x) and ψ(x) defined below are known as the Chebyshev functions. The function θ(x) is defined by θ(x) | congruent | sum_(k = 1)^(π(x)) ln p_k | = | ln[ product_(k = 1)^(π(x)) p_k] | = | ln x# (Hardy and Wright 1979, p. 340), where p_k is the kth prime, π(x) is the prime counting function, and x# is the primorial.
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