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

DOI:
https://doi.org/10.1090/S0025-5718-1984-0726005-8

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