For any > 0 and any non-exceptional modulus q â‰¥ 3, we prove that, for x large enough (x â‰¥ Î± log q), the interval [ ex, ex+ ] contains a prime p in any of the arithmetic progressions modulo q. Weâ€¦ (More)

N(Ïƒ, T ) â‰¤ 4.9(3T ) 8 3 (1âˆ’Ïƒ) log5âˆ’2Ïƒ(T ) + 51.5 log T, for Ïƒ â‰¥ 0.52 and T â‰¥ 2000. We discuss a generalization of the method used in these two results which yields an explicit bound of a similarâ€¦ (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 Î¶K(s) has no zeros in the region:â€¦ (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 eâ†µective constant computed in terms of x0.

Robots are playing a vital role in todayâ€™s industrial automation and monitoring system. As technology developed these robots have increased their applications and functionality. Working robots willâ€¦ (More)