Explicit estimates for 𝜃(𝑥;3,𝑙) and 𝜓(𝑥;3,𝑙)

McCurley, Kevin S.

doi:10.1090/S0025-5718-1984-0726005-8

Explicit estimates for $\theta (x;3,l)$ and $\psi (x;3,l)$
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Math. Comp. 42 (1984), 287-296 Request permission

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 $|\theta (x;3,l) - x/2| < ex$, $x \geqslant {x_0}(\varepsilon )$.

References

Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2
D. Davies, An approximate functional equation for Dirichlet $L$-functions, Proc. Roy. Soc. London Ser. A 284 (1965), 224–236. MR 173352, DOI 10.1098/rspa.1965.0060
Kevin S. McCurley, Explicit zero-free regions for Dirichlet $L$-functions, J. Number Theory 19 (1984), no. 1, 7–32. MR 751161, DOI 10.1016/0022-314X(84)90089-1
Kevin S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004, DOI 10.1090/S0025-5718-1984-0726004-6
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
Robert Spira, Calculation of Dirichlet $L$-functions, Math. Comp. 23 (1969), 489–497. MR 247742, DOI 10.1090/S0025-5718-1969-0247742-X