This paper introduces the Vadraj Series, a parameter-dependent complex-valued infinite sum that couples the M¨obius function μ(n) with a cosine oscillator whose frequency is driven by the square roots of the prime numbers, √pn, scaled by an integer parameter k and a continuous regulator ε. We derive the exact algebraic reduction of the series’ complex denominator to polar form, and prove that the oscillatory phase driving the series has no limit as ε → 0+. We state precisely what is and is not established by this result. We present a reproducible numerical exploration of the series’ partial sums, and we discuss — as motivation only, not as a proven equivalence — the series’ relation to the growth rate of the Mertens function and, through it, to the Riemann Hypothesis
Brijesh Vadraj (Sun,) studied this question.