The date of the European trip drew nearer; the lecture intended to elucidate the bridge‑role of the Langlands program had been refined to near‑perfection with the aid of the virtual‑reality system—every metaphor, every visualization link meticulously polished. Yet deep within Yue'er's heart, a more profound gravitational pull was drawing her away from expounding the macro‑framework toward a more abyssal, more fundamental mathematical chasm. The most critical load‑bearing structure of this grand bridge connecting number theory and geometry was not those seemingly splendid correspondences themselves, but the ultimate secret hidden beneath—concerning the very stability of the bridge: **the distribution of zeros of L‑functions.** To touch this secret, one could not bypass that majestic snow‑capped mountain standing on the mathematical horizon for over a century and a half, where countless geniuses had met their defeat: the **Riemann Hypothesis.**
Yue'er's study seemed transformed into a gravitational anomaly point. Spread on the desk were no longer merely literature on the Langlands program and her own lecture drafts, but more so weighty classics and cutting‑edge papers on the Riemann zeta‑function, analytic number theory, and the theory of more general L‑functions. The air was permeated with a stillness more distilled, closer to the source of mathematics itself, than during speech preparation. Her gaze pierced through the dazzling surface of Langlands correspondences, pointing straight at the beating heart—those mysterious **non‑trivial zeros** that determine the properties of L‑functions and thus whether the correspondences can hold.
She sought a metaphor, for herself and for potential future listeners, to grasp its importance. Finally, an image from modern cosmology surfaced in her mind: **cosmic microwave background radiation.**
The afterglow of the Big Bang, uniformly permeating the entire universe, appears bland and unremarkable; yet physicists know it contains the entire code for the formation of early‑universe structure. Those tiny, one‑hundred‑thousandth‑level temperature fluctuations, like fingerprints imprinted on the spacetime background, reveal all initial conditions for how matter condensed from uniformity to form galaxies, stars, even life.
In her view, **the distribution of non‑trivial zeros of L‑functions** was the "background radiation" of the mathematical universe—the "hidden code" revealing its deepest structures and symmetries.
An L‑function, whether originating from a number‑theory object (like an algebraic equation) or a geometric object (like an automorphic form), when analytically continued to the whole complex plane, contains explosive information about the nature of that object at points where its value equals zero—its "zeros." Zeros fall into two categories: "trivial zeros," whose positions are directly determined by the functional equation of the function, like the known, smooth baseline in cosmic background radiation; and the truly crucial ones—the **"non‑trivial zeros"**—whose positions appear random yet follow some profound, not‑fully‑understood internal law.
The Riemann Hypothesis, targeting the most famous L‑function—the Riemann zeta‑function—makes an extraordinarily bold and elegant assertion: **All non‑trivial zeros lie on the vertical line with real part 1/2 in the complex plane (the critical line).** This line is therefore called the "critical line." If one imagines the Riemann zeta‑function as a complex "mathematical terrain," then its "coastline" where elevation is zero is conjectured to consist solely of this straight "1/2 line."
The profound significance of this conjecture shook Yue'er's soul:
**The ultimate law of prime distribution:** The non‑trivial zeros of the Riemann zeta‑function stand in a precise duality with the distribution of primes among natural numbers, akin to a Fourier transform. If the Riemann Hypothesis holds, it means prime distribution is not completely chaotic; rather, it is regulated in an extremely precise manner by that "1/2 line" hidden deep in the complex plane. We could obtain the strongest possible estimates for the size of the n‑th prime, and optimal error terms for primes within intervals. This would be an ultimate leap in humanity's understanding of arithmetic elementary units. **The epitome of mathematical symmetry:** All non‑trivial zeros neatly aligned on a single line—this itself is an astonishing, utmost mathematical symmetry and beauty. It hints that at the foundation of the mathematical universe, there exists a powerful "selectiveness" or "forbidden rule," not yet fully understood, which allows zeros to appear only at this specific "energy level." This symmetry may be more fundamental and profound than any symmetry we currently know. **The cornerstone of the Langlands program:** In the Langlands program, proving that two objects (e.g., a Galois representation and an automorphic form) correspond hinges on proving that their L‑functions are identical. And a key (often final) step in proving two L‑functions identical is to show they possess **exactly the same** set of non‑trivial zeros (including positions and multiplicities). The distribution of zeros becomes the "DNA fingerprint" verifying whether a correspondence holds. The highly ordered zero distribution predicted by the Riemann Hypothesis, if proven to be a universal law applicable to a large class of L‑functions (the Generalized Riemann Hypothesis), would provide a potential, unified proof path for countless specific correspondences in the Langlands program, greatly simplifying and deepening our understanding of mathematical unity.
Yue'er immersed herself in this abstract world of complex planes, infinite series, and hidden zeros. She read about algorithms for computing large‑scale zeros, pondered possible approaches to proving zeros lie on the critical line, felt the intellectual tremor of attempting to capture the universe's deepest code. The high abstraction of this field kept it remote from any direct technological application, even from the daily research scope of most mathematicians. It was pure, profound, yet… exceptionally lonely.
Sometimes, late at night, when she lifted her head from a pile of drafts covered with Σ integral signs and complex‑function graphs, feeling her mind somewhat drained from extreme concentration, an indescribable sense of alienation would steal over her. She would recall Mozi battling invisible titans in the volatile waves of capital markets, a struggle concerning immense funds changing in split seconds; she would recall Xiuxiu in the cleanroom, waging arduous trench warfare against nanometer‑scale water flow, bubbles, and physical defects—a battle concerning physical manufacturing.
The world they inhabited was a "real world" filled with tangible tension, immediate feedback, and concrete results. Capital growth, stable operation of a lithography machine, successful tape‑out of a chip—these outcomes were visible, quantifiable, their value seemingly instant.
And herself? The points on the complex plane she obsessed over, these conjectures about infinity and symmetry, this seemingly ethereal "background code of the universe"—what was their significance? Beyond satisfying the pure curiosity of a handful of scholars, what contribution did they make to that "real world" driven by capital and lithography machines? This stark contrast would occasionally invade her fervent research core like a chill, causing a momentary daze and self‑doubt. Was she living too "metaphysically," already disconnected from those comrades who were truly shaping and changing the world?
During a routine encrypted video call, Xiuxiu excitedly reported to her and Mozi the major breakthrough of the immersion water film remaining stable for ten minutes, showing photos of the team weeping with joy. Mozi also calmly analyzed the capital growth brought by the trend model's great victory in the gold market, and the more covert layouts of hidden opponents. When it was Yue'er's turn, she opened her mouth, wanting to share her thoughts on L‑function zeros and the Riemann Hypothesis, only to find those concepts so abstract that, compared to "water film stable for ten minutes" and "capital growth by billions," they seemed so… detached from worldly affairs.
She finally just said, "I'm preparing some deep mathematical foundations for the lecture, about… the zeros of functions." Her tone inadvertently carried a touch of despondency that even she hadn't noticed.
Mozi keenly caught the subtle emotion in her voice. He didn't press for specific mathematical details; instead, after Xiuxiu finished sharing, he looked at the somewhat silent Yue'er on the screen and spoke calmly:
"Yue'er," his voice came through the ether, steady and forceful, "Xiuxiu stabilized the water film—that is laying the **physical cornerstone** for chip manufacturing; I operate capital—that is providing the **energetic cornerstone** for all these explorations and manufacturing. And you—"
He paused, his gaze seeming to penetrate the screen, looking straight into her heart. "What you explore is the **cornerstone beneath all these cornerstones**. The cornerstone of logic, the cornerstone of law, the **first‑principles** of understanding why this world can be driven by capital and carved by light."
"Without your exploration of 'strings,' my 'code' might never touch the true 'soul'; without your insight into mathematical unity, Xiuxiu's 'light' might never find the most efficient, most essential path. The 'zeros' you study now, these 'conjectures'—to many they may seem unreachable, but they represent the ultimate ambition of human intelligence to understand the underlying operating rules of the universe. That ambition itself is the most precious wealth, the **source spring** of all applications and technology."
His tone was firm, leaving no room for doubt. "Do not doubt the value of your path. Your world is equally real—more fundamental, more eternal reality. The three of us, missing any cornerstone, the entire edifice would collapse. Your exploration is equally crucial."
Yue'er was stunned; that faint chill in her heart instantly melted away in the warm current of Mozi's words. She looked at Mozi's resolute eyes on the screen, and Xiuxiu's vigorous nodding in agreement beside him; a warmth flooded her entire being.
Yes, capital, technology, theory—the three formed a stable triangle. Mozi guarded the boundaries of reality, Xiuxiu expanded the frontiers of matter, and she explored the source of logic and law. They were not disconnected; they were at different levels of the same grand narrative, interdependent, mutually nourishing.
She took a deep breath; her eyes regained clear, firm light. "I understand," she said softly. That mathematical starry sky, occasionally lonely due to high abstraction, was once again illuminated by her comrades' understanding and affirmation. The snow‑capped mountain of the Riemann Hypothesis remained majestic, the zeros of L‑functions remained mysterious, but she knew her climb, her interpretation, were equally indispensable for the future they three together were building—that future called "String Light." She no longer felt lonely, no longer doubted. She would continue deeper into that mathematical universe's background radiation composed of zeros and conjectures, to decipher the ultimate code hidden deepest within—about order and unity.