Inspiration often came when least expected.
Earlier, while teaching his class, Lu Zhou had reviewed the history of the Riemann Hypothesis and recalled many details he had overlooked while researching the problem.
Most of the time, those details were inconsequential even when he remembered them. At a critical moment, however, even inconsequential clues could play a surprisingly important role.
After returning to his office, Lu Zhou sat down at his desk, took some scratch paper out of a drawer, and wrote down his earlier thoughts line by line.
"The Riemann Hypothesis was proposed to study the prime-counting function, π(x). Riemann's pioneering contribution wasn't proposing the Riemann Hypothesis itself, but introducing the Riemann zeta function and taking the study of π(x) from the real line to the complex plane."
His pen glided lightly across the paper as he smoothly wrote down an equation.
[π(x)=∫dtlnt+O(x^(12+ε))...]
"An equivalent formulation of the Riemann Hypothesis..." He stopped writing and tapped the paper lightly with the tip of his pen. His expression gradually grew solemn with concentration as he murmured thoughtfully, "If the Riemann Hypothesis holds, this asymptotic formula also holds. Conversely, this asymptotic formula can also imply the Riemann Hypothesis..."
"The key to solving the problem should lie in π(x) itself!"
An inkling of understanding gradually formed in his mind, but he still felt something was missing.
It was like a jigsaw puzzle that was already a third complete, with scattered pieces lying beside it. Nothing seemed to be missing, except the single most crucial key.
Finding it would allow him to connect all the clues. Without it, everything seemed to be nothing more than empty speculation.
He sat at his desk like that for an entire hour. Only when his stomach gave a quiet growl did Lu Zhou abruptly come back to his senses.