• 대한수학회지
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• 제59권5호
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• pp.843-867
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• 2022

Maynard proved that there exists an effectively computable constant q1 such that if q ≥ q1, then $\frac{{\log}\;q}{\sqrt{q}{\phi}(q)}Li(x){\ll}{\pi}(x;\;q,\;m)<\frac{2}{{\phi}(q)}Li(x)$ for x ≥ q8. In this paper, we will show the following. Let 𝛿1 and 𝛿2 be positive constants with 0 < 𝛿1, 𝛿2 < 1 and 𝛿1 + 𝛿2 > 1. Assume that L ≠ ℚ is a number field. Then there exist effectively computable constants c0 and d1 such that for dL ≥ d1 and x ≥ exp (326n𝛿1L(log dL)1+𝛿2), we have $$\|{\pi}_C(x)-\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)\|\;{\leq}\;\(1-c_0\frac{1og\;d_L}{d^{7.072}_L}\)\;\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)$$.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.37-51
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• 2003

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function ,r2,4(5${\times}$10$\^$10/) = l18905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computed B$_4$(5 ${\times}$ 10$\^$10/) = 1.1970s4473029 and B$_4$ = 1.197054 ${\pm}$ 7 ${\times}$ 10$\^$-6/. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function $\pi$$\_$2.6/(5${\times}$10$\^$10/) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computed B$\_$6/(5${\times}$10$\^$10/) = 0.93087506039231 and B$\_$6/ = 1.135835 ${\pm}$ 1.2${\times}$10$\^$-6/.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.441-452
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• 2006

We introduce the counting function ${\pi}^*_{2.8}(x)$ of the primes with difference 8 between consecutive primes ($p_n,\;p_{n+l}=p_n+8$) can be approximated by logarithm integral $Li^*_{2.8}$. We calculate the values of ${\pi}^*_{2.8}(x)$ and the sum $C_{2,8}(x)$ of reciprocals of primes with difference 8 between consecutive primes $p_n,\;p_{n+l}=p_n+8$ where x is counted up to $7{\times}10^{10}$. From the results of these calculations. we obtain ${\pi}^*_{2.8}(7{\times}10^{10}$)= 133295081 and $C_{2.8}(7{\times}10^{10}) = 0.3374{\pm}2.6{\times}10^{-4}$.