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Mathematics of Computation

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Explicit estimates for $\theta (x;3,l)$ and $\psi (x;3,l)$

Author: Kevin S. McCurley
Journal: Math. Comp. 42 (1984), 287-296
MSC: Primary 11N56
MathSciNet review: 726005
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Abstract: Let $\theta (x;3,l)$ be the sum of the logarithms of the primes not exceeding x that are congruent to l modulo 3, where l is 1 or 2. By the prime number theorem for arithmetic progressions, $\theta (x;3,l) \sim x/2$ as $x \to \infty$. Using information concerning zeros of Dirichlet L-functions, we prove explicit numerical bounds for $\theta (x;3,l)$ of the form $|\theta (x;3,l) - x/2| < ex$, $x \geqslant {x_0}(\varepsilon )$.

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Article copyright: © Copyright 1984 American Mathematical Society