Zeros of continuous functions and the Mahler measure: an exact integral identity

We establish an exact complex identity linking, in a single integral, the position of the zero x∗x^* x∗ of a continuous function ff f and its Mahler measure M(f) The proof is three lines. The real part encodes the Mahler measure, the imaginary part encodes the zero via the Tope formula. Validated numerically to 10−5010⁻⁵⁰ 10−50 on eight test functions. An application to the Riemann zeta function is given.