The Extended Riemann Hypothesis and its Application to Computation

Many of Hilbert’s 23 famous problems are not of a prove or disprove nature; rather, they are open-ended, “of a purely investigative nature,” [12] and may never be answered to satisfaction. One of the best examples of this is the sixth problem, that of the axiomatization of physics. Hilbert said, “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics” [15]. Eight of the 23 problems can be regarded this way, and of the other 15, just three remain unsolved. One of those three, the Riemann Hypothesis, has remained “as mysterious and challenging as ever” [12] and is thus one the Clay Mathematics Institute’s seven Millennium Prize Problems. The purpose of this paper is to survey the uses of the Extended Riemann Hypothesis in creating algorithms to perform various mathematical tasks. We will begin with an introduction to the Riemann Hypothesis and Extended Riemann Hypothesis, then move on to the applications. We will only spend time on proving one of those applications (the Solovay-Strassen primality test, which will be our first topic), but we will also discuss factoring polynomials over finite fields, factoring integers, searching for primitive roots, and finding k power nonresidues modulo a prime. We will conclude with a brief discussion of why the Extended Riemann Hypothesis might be true.