There were many mathematicians who tried to prove or disprove the statement of Riemann Hypothesis. However, none of them have been successfully approved by the Clay Mathematical Institute. In addition, to the best of this author’s knowledge, these mathematicians haven’t employed the technique of logical truth table during their proofs. With reference to this author’s previous proof in [1], this author have employed the method of multiplicative telescope together with the prime boundary gaps. In this extended version of my proof to the Riemann Hypothesis, this author tries to show that RH statement is true through the four cases of the conditional statements in the truth table. Three of the cases (I, II, IV) are found to be true for the conditional statement in the Riemann Hypothesis while only one (case III) is found to be false (and acts as the disproof by a counter-example). Moreover, there are also three sub-cases (i, ii, iii) [1] among these four tabled cases. The main idea is that the we may disproof the hypothesis statement that is similar to the RH one by first find a counter-example which is obviously a disproof (case III) to the (Riemann) hypothesis. But it is NOT compatible with the GÖdel’s Incompleteness Theorem. Otherwise either the disproof to the statement or the Gödel is incorrect which is impossible. Hence, the disproof is said to be incompatible with the Gödel. On the other hand, all of the other truth cases (I, II, IV) for the statement are indeed the examples for the positive results to the Riemann Hypothesis statement and are compatible with the Gödel. Therefore, the only way to make a conclusion is to say or force the Riemann Hypothesis statement to be correct.In general, for any hypothesis with the conditional statements structure like the Riemann one, we may also prove them by the similar techniqe and the arguments of the truth table for their conditional statements together with the Gödel’s Incompleteness theorem to force the positive result for the hypothesis statement. Actually, there are many applications for the truth tables especially in the fields like language (structure & modeling) or in engineering (logic gates & programming) etc during our everyday usage.
Keywords: extension, logical entails truth table, proof of Riemann hypothesis
