The Zetarc constant (typically denoted by the symbol ${\displaystyle Z_{a}}$) is a mathematical constant defined as the sine of the square root of the Riemann zeta function evaluated at 2 [3]. Formally expressed as: ${\displaystyle Z_{a}=\sin(\surd ({\frac {\pi }{3}})\times ({\frac {\pi }{2}}))}$

The constant is a transcendental number with an approximate decimal value of 0.9587438202539251...

The term is a portmanteau of Zeta (referring to the Riemann zeta function) and arc (evoking trigonometric arc and wave properties, though operating as a forward sine function).

The Zetarc constant was named to reflect its structural reliance on both analytic number theory and classical trigonometry.

By utilizing Euler's solution to the Basel problem, the value of ζ(2) is known to be exactly equal to ${\displaystyle {\frac {\pi ^{2}}{6}}}$. Therefore, the constant can be evaluated through three distinct mathematical phases:

1. The Core Input: The Riemann zeta value ${\displaystyle \zeta (2)={\frac {\pi ^{2}}{6}}\approx 1.644934\ldots }$
2. The Radian Argument: The square root of the input ${\displaystyle {\frac {\pi }{\surd 6}}\approx 1.2825498}$ ${\displaystyle {\textrm {radians}}}$
3. The Trigonometric Mapping: The sine of the resulting radians, yielding the final constant ≈ 0.95874382

The decimal expansion of the Zetarc constant begins as follows: ${\displaystyle Z_{a}=0.9587438202539251...}$

The term is a portmanteau of Zeta (referring to the Riemann zeta function) and arc (evoking trigonometric arc and wave properties, though operating as a forward sine function).

Because π is a transcendental number (it is not the root of any non-zero polynomial with rational coefficients), both  ${\displaystyle {\frac {\pi }{\surd 6}}}$ and its sine, ${\displaystyle Z_{a}}$ are transcendental numbers.

In a unit circle, if an arc length of exactly ${\displaystyle {\frac {\pi }{\surd 6}}}$ units (roughly 73.48°) is measured along the circumference starting from (1,0), the y-coordinate of the resulting point on the circle is exactly equal to the Zetarc constant.