Explicit estimates for and

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 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, as . Using information concerning zeros of Dirichlet *L*-functions, we prove explicit numerical bounds for of the form , .

**[1]**R. Brent, "On the zeros of the Riemann zeta function in the critical strip,"*Math. Comp.*, v. 33, 1979, pp. 1361-1372. MR**537983 (80g:10033)****[2]**D. Davies, "An approximate functional equation for Dirichlet*L*-functions,"*Proc. Roy. Soc. London*, v. 284, 1965, pp. 224-236. MR**0173352 (30:3565)****[3]**K. S. McCurley, "Explicit zero-free regions for Dirichlet*L*-functions." (To appear.) MR**751161 (85k:11041)****[4]**K. S. McCurley, "Explicit estimates for the error term in the prime number theorem for arithmetic progressions,"*Math. Comp.*, v. 42, 1984, pp. 265-285. MR**726004 (85e:11065)****[5]**J. B. Rosser & L. Schoenfeld, "Sharper bounds for the Chebyshev functions and ,"*Math. Comp.*, v. 29, 1975, pp. 243-269. MR**0457373 (56:15581a)****[6]**Royal Society of London, Mathematical Tables Committee, Royal Society Depository of Unpublished Mathematical Tables, Table 83.**[7]**L. Schoenfeld, "Sharper bounds for the Chebyshev functions and . II,"*Math. Comp.*, v. 30, 1976, pp. 337-360. MR**0457374 (56:15581b)****[8]**R. Spira, "Calculation of Dirichlet L-functions,"*Math. Comp.*, v. 23, 1969, pp. 489-497. Microfiche supplement. MR**0247742 (40:1004a)**

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

Article copyright:
© Copyright 1984
American Mathematical Society