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For any > 0 and any non-exceptional modulus q ≥ 3, we prove that, for x large enough (x ≥ α log q), the interval [ ex, ex+ ] contains a prime p in any of the arithmetic progressions modulo q. We… (More)
N(σ, T ) ≤ 4.9(3T ) 8 3 (1−σ) log5−2σ(T ) + 51.5 log T, for σ ≥ 0.52 and T ≥ 2000. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar… (More)
Let K be a number field, nK its degree, and dK the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region:… (More)
We prove that if x is large enough, namely x x0, then there exists a prime between x(1 1) and x, where is an e↵ective constant computed in terms of x0.
Robots are playing a vital role in today’s industrial automation and monitoring system. As technology developed these robots have increased their applications and functionality. Working robots will… (More)
Let Lq(s) be the product of Dirichlet L-functions modulo q. Then Lq(s) has at most one zero in the region Rs ≥ 1 − 1 6.3970 logmax(q, q|Is|) .