- \(^\ast\) One of the “explicit formulas” in the theory of primes is as follows: if \(\psi _ { 1 }\) is the integrated Tchebychev function considered in Section 2, then
\[ \psi_ {1} (x) = \frac {x ^ {2}}{2} - \sum_ {\rho} \frac {x ^ {\rho}}{\rho (\rho + 1)} - E (x) \]
where the sum is taken over all zeros \(\rho\) of the zeta function in the critical strip. The error term is given by \(E ( x ) = c _ { 1 } x + c _ { 0 } + \sum _ { k = 1 } ^ { \infty } x ^ { 1 - 2 k } / ( 2 k ( 2 k - 1 ) )\) , where \(c _ { 1 } = \zeta ^ { \prime } ( 0 ) / \zeta ( 0 )\) and \(c _ { 0 } = \zeta ^ { \prime } ( - 1 ) / \zeta ( - 1 )\) . Note that \(\textstyle \sum _ { \rho } 1 / | \rho | ^ { 1 + \epsilon } < \infty\) for every \(\epsilon > 0\) , because \(( 1 - s ) \zeta ( s )\) has order of growth 1. (See Exercise 8.) Also, obviously \(E ( x ) = O ( x )\) as \(x \to \infty\)