Let 1/2+ia_{k} be the k-th non-trivial zeros of the Riemann Zeta function. I have the following conjectures. Please let me known what is known towards this.
Conjecture1: There exists infinitely many positive integers k such the the integral parts [a_k]=[a_{k+1}], where 1/2+ia_{k} be the k-th non-trivial zeros of the Riemann Zeta function.
Conjecture2: For any given integer n, there exists infinitely many positive integers k such the the integral parts [a_k]=[a_{k+1}]=...=[a_{k+n-1}], where 1/2+ia_{k} be the k-th non-trivial zeros of the Riemann Zeta function.
In particular, Conjecture 1 is a special case of conjecture2 when n=2.
The value of any conjecture is what it leads to if it is true-or if it is false. So you have to show what would follow, were this conjecture true, what would follow were it false and why either consequence would be of any interest in understanding the properties of the zeta function. it's always possible to make ``what if?" claims; they need to provide answers to the question ``so what?".
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