for almost two hours)—there
is a knock on the door. At another time in his life, this wouldn't have surprised him. A visitor at three in the morning would
have been routine. Now, though, the knock disorients him, throws him into a panic. "Just a minute," he calls, putting on his
dressing gown. "Who is it?"
"Me. Littlewood."
He opens the door. Littlewood strides in, dripping and umbrellaless.
"The stuff about primes is wrong," he says.
"What?"
"Oh, sorry. Have I woken you?"
"It doesn't matter. Come in."
Without even removing his coat, Littlewood heads for the blackboard, still covered with Hardy's earlier scribblings. "I couldn't
sleep so I started looking over the letter and—may I?"
"Of course."
"Right, so this is what I think he's done." He wipes the blackboard clean. "Here's his formula for calculating the number
of primes less than n. Well, it's the usual Riemannian formula, except that he's left out the terms coming from the zeros of the zeta function. And
his results—I've tested them—are just what you'd get if the zeta function had no non-trivial zeros."
"Damn."
"I have a vague theory as to how the mistake came about. He's staking everything on the legitimacy of some operations he's
doing on divergent series, banking on a hunch that if the first results are correct, the theorem must be true. And the first
results are correct. Even up to a thousand the formula gives exactly the right answer. Unfortunately, he had no one around to warn him
that the primes like to misbehave once they get larger."
"Still, leaving out the zeros . . . it's not an encouraging sign."
"Oh, but there I disagree, Hardy. I think it's a very encouraging sign." Littlewood steps closer. "You must realize, ordinary
mathematicians don't make mistakes like this. Even very good mathematicians don't make mistakes like this. And when you consider
the other stuff, the stuff on continued fractions and elliptic functions . . . I can believe he's at least a Jacobi."
Hardy raises his eyebrows. Now this is high praise. Since starting at Trinity, he's kept up a mental ranking of the great mathematicians, putting each in the
class of a cricketer whom he admires. He judges himself the equal of Shrimp Levison-Gower, Littlewood on a par with Fry, Gauss
in the category of Grace, the greatest player in the history of the game. Jacobi, the last time Hardy ranked him, was somewhere
above Fry but below Grace—in the vicinity of the young and dazzling Jack Hobbs—which means that Ramanujan, if Littlewood is
correct, might have the potential to be another Grace. He could well prove the Riemann hypothesis.
"What about the rest?"
"I haven't had a chance to go over these other asymptotic formulas, but at first glance, they look to be totally original.
And significant."
"But no proofs."
"I don't think he really understands what a proof is, or that it's important to give one, because he's been working on his
own all these years, and who knows what books, if any, he has access to. Perhaps no one's taught him. Could you teach him?"
"I've never tried to teach anyone why you have to give proofs. My students have always just . . . understood."
There is a moment of quiet, now, of which Hermione takes advantage by rubbing herself against Littlewood's leg. When he tries
to pick her up, she runs for cover beneath the ottoman.
"A tease, that female. Come here, kitten!"
"She can't hear you. Remember?"
"Oh, of course." Littlewood regards the floor.
"And what are we to do now?" Hardy asks.
"Is there any question? Bring him to England."
"He's said nothing about wanting to come."
"Of course he wants to come. Why else would he have written? And what's he got in Madras? A clerkship."
"But if we get him here, will we know what to do with him?"
"I think the more apt question is, will he know what to do with us?"
Littlewood pushes his glasses up on his nose. "Have you heard from the India Office, by the way?"
"Not
is a knock on the door. At another time in his life, this wouldn't have surprised him. A visitor at three in the morning would
have been routine. Now, though, the knock disorients him, throws him into a panic. "Just a minute," he calls, putting on his
dressing gown. "Who is it?"
"Me. Littlewood."
He opens the door. Littlewood strides in, dripping and umbrellaless.
"The stuff about primes is wrong," he says.
"What?"
"Oh, sorry. Have I woken you?"
"It doesn't matter. Come in."
Without even removing his coat, Littlewood heads for the blackboard, still covered with Hardy's earlier scribblings. "I couldn't
sleep so I started looking over the letter and—may I?"
"Of course."
"Right, so this is what I think he's done." He wipes the blackboard clean. "Here's his formula for calculating the number
of primes less than n. Well, it's the usual Riemannian formula, except that he's left out the terms coming from the zeros of the zeta function. And
his results—I've tested them—are just what you'd get if the zeta function had no non-trivial zeros."
"Damn."
"I have a vague theory as to how the mistake came about. He's staking everything on the legitimacy of some operations he's
doing on divergent series, banking on a hunch that if the first results are correct, the theorem must be true. And the first
results are correct. Even up to a thousand the formula gives exactly the right answer. Unfortunately, he had no one around to warn him
that the primes like to misbehave once they get larger."
"Still, leaving out the zeros . . . it's not an encouraging sign."
"Oh, but there I disagree, Hardy. I think it's a very encouraging sign." Littlewood steps closer. "You must realize, ordinary
mathematicians don't make mistakes like this. Even very good mathematicians don't make mistakes like this. And when you consider
the other stuff, the stuff on continued fractions and elliptic functions . . . I can believe he's at least a Jacobi."
Hardy raises his eyebrows. Now this is high praise. Since starting at Trinity, he's kept up a mental ranking of the great mathematicians, putting each in the
class of a cricketer whom he admires. He judges himself the equal of Shrimp Levison-Gower, Littlewood on a par with Fry, Gauss
in the category of Grace, the greatest player in the history of the game. Jacobi, the last time Hardy ranked him, was somewhere
above Fry but below Grace—in the vicinity of the young and dazzling Jack Hobbs—which means that Ramanujan, if Littlewood is
correct, might have the potential to be another Grace. He could well prove the Riemann hypothesis.
"What about the rest?"
"I haven't had a chance to go over these other asymptotic formulas, but at first glance, they look to be totally original.
And significant."
"But no proofs."
"I don't think he really understands what a proof is, or that it's important to give one, because he's been working on his
own all these years, and who knows what books, if any, he has access to. Perhaps no one's taught him. Could you teach him?"
"I've never tried to teach anyone why you have to give proofs. My students have always just . . . understood."
There is a moment of quiet, now, of which Hermione takes advantage by rubbing herself against Littlewood's leg. When he tries
to pick her up, she runs for cover beneath the ottoman.
"A tease, that female. Come here, kitten!"
"She can't hear you. Remember?"
"Oh, of course." Littlewood regards the floor.
"And what are we to do now?" Hardy asks.
"Is there any question? Bring him to England."
"He's said nothing about wanting to come."
"Of course he wants to come. Why else would he have written? And what's he got in Madras? A clerkship."
"But if we get him here, will we know what to do with him?"
"I think the more apt question is, will he know what to do with us?"
Littlewood pushes his glasses up on his nose. "Have you heard from the India Office, by the way?"
"Not
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