The analytic theory of numbers was born in 1837, with Dirichlet's striking proof of the infinitude of primes in arithmetic progressions. Yet it didn't reach its peak until the end of the century. Some years before Dirichlet, Carl Friedrich Gauss had arrived at a good guess of an 'asymptotic' formula (i.e. an approximation, getting better and better as n grows) of the number of primes less than a certain integer n. Yet neither he nor anyone after him had been able to suggest a hint of a proof. Then in 1859, Bernhard Riemann introduced an infinite sum in the plane of complex numbers, [8] ever since known as the 'Riemann Zeta Function', which promised to be an extremely useful new tool. To use it effectively, however, number theorists had to abandon their traditional, algebraic (so-called 'elementary') techniques and resort to the methods of Complex Analysis, i.e. the infinitesimal calculus applied to the plane of complex numbers.
A few decades later, when Hadamard and de la Vallee-Poussin managed to prove Gauss's asymptotic formula using the Riemann Zeta Function (a result henceforth known as the Prime Number Theorem) the analytic approach suddenly seemed to become the magic key to the innermost secrets of Number Theory.
It was at the time of this high tide of hope in the analytic approach that Petros began his work on Goldbach's Conjecture.
After spending the initial few months familiarizing himself with the dimensions of his problem, he decided he would proceed through the Theory of Partitions (the different ways of writing an integer as a sum), another application of the analytic method. Apart from the central theorem in the field, by Hardy and Ramanujan, there also existed a hypothesis by the latter (another of his famous 'hunches') which Petros hoped would become a crucial stepping stone to the Conjecture itself – if only he managed to prove it.
He wrote to Littlewood, asking as discreetly as possible whether there had been any more recent developments in this matter, his question purportedly expressing 'a colleague's interest'. Littlewood reported in the negative, also sending him Hardy's new book, Some Famous Problems of Number Theory. In it, there was a proof of sorts of what is known as the Second or 'other' Conjecture of Goldbach [9]. This so-called proof, however, had a fundamental lacuna: its validity relied on the (unproven) Riemann Hypothesis. Petros read this and smiled a superior smile. Hardy was becoming pretty desperate, publishing results based on unproven premises! Goldbach's main Conjecture, the Conjecture, as far as he was concerned, was not even given lip service; his problem was safe.
Petros conducted his research in total secrecy, and the deeper his probing led him into the terra incognita defined by the Conjecture, the more zealously he covered his tracks. For his more curious colleagues he had the same decoy answer that he'd used with Hardy and Littlewood: he was building on the work he had done with them in Cambridge, continuing their joint research on the Riemann Hypothesis. With time, he became cautious to the point of paranoia. In order to avoid his colleagues' drawing conclusions from the items he withdrew from the library, he began to find ways of disguising his requests. He would protect the book he really wanted by including it in a list of three or four irrelevant ones, or he would ask for an article in a scientific journal only in order to get his hands on the issue that also contained another article, the one he really wanted, to be perused far from inquiring eyes in the total privacy of his study.
In the spring of that year, Petros received an additional short communication from Hardy, announcing Srinivasa Ramanujan's death of tuberculosis, at the age of thirty-two, in a slum neighbourhood of Madras. His first reaction to the sad news perplexed and even distressed him. Under a surface layer of sorrow for the loss of the extraordinary mathematician and the gentle, humble, sweet-spoken friend, Petros feit deep inside a wild joy that this phenomenal brain was no longer in the arena of Number Theory.
You see, he had feared no one else. His two most qualified rivals, Hardy and Littlewood, were too involved with the Riemann Hypothesis to think seriously about Goldbach's Conjecture. As to David Hubert, generally acknowledged to be the world's greatest living mathematician, or Jacques Hadamard, the only other number theorist to be reckoned with, both were by now really no more than esteemed veter-ans – their almost sixty years were tantamount to advanced old age for creative mathematicians. But he had feared Ramanujan. His unique intellect was the only force he considered capable of purloining his prize. Despite the doubts he had expressed to Petros about the general validity of the Conjecture, should Ramanujan ever have decided to focus his genius on the problem… Who knows, maybe he would have been able to prove it despite himself; maybe his dear goddess Namakiri would have offered the solution to him in a dream, all neatly written out in Sanskrit on a roll of parchment!
Now, with his death, there was no longer any real danger of someone arriving at the solution before Petros.
Still, when he was invited by the great School of Mathematics at Göttingen to deliver a memorial lecture on Ramanujan's contribution to Number Theory, he carefully avoided mentioning his work on Partitions, lest anyone be inspired to look into its possible connections with Goldbach's Conjecture.
In the late summer of 1922 (as it happened, on the very same day that his country was ravaged by the news of the destruction of Smyrna) Petras was suddenly faced with his first great dilemma.
The occasion was a particularly happy one: while taking a long walk on the shore of the Speichersee, he arrived by way of a sudden illumination, following months of excruciating work, at an amazing insight. He sat down in a small beer-garden and scribbled it in the notebook he always carried with him. Then he took the first train back to Munich and spent the hours of dusk till dawn at his desk, working out the details and going over his syllogism carefully, again and again. When he was finished he felt for the second time in his life (the first had to do with Isolde) a feeling of total fulfilment, absolute happiness. He had managed to prove Ramanujan's hypothesis!
In the first years of his work on the Conjecture, he had accumulated quite a few interesting intermediate results, so-called 'lemmas' or smaller theorems, some of which were of unquestionable interest, ample material for several worthwhile publications. Yet he had never been seriously tempted to make these public. Although they were respectable enough, none of them could qualify as an important discovery, even by the esoteric standards of the number theorist.
But now things were different.
The problem he had solved on his afternoon walk by the Speichersee was of particular importance. As regarded his work on the Conjecture it was of course still an intermediate step, not his ultimate goal. Nevertheless, it was a deep, pioneering theorem in its own right, one which opened new vistas in the Theory of Numbers. It shed a new light on the question of Partitions, applying the previous Hardy-Ramanujan theorem in a way that no one had suspected, let alone demonstrated, before. Undoubtedly, its publication would secure him recognition in the mathematical world much greater than that achieved by his method for solving differential equations. In fact, it would probably catapult him to the first ranks of the small but select international Community of number theorists, practically on the same level as its great stars, Hadamard, Hardy and Littlewood.