Denote by $\lambda(n)$, the number of numbers between $0$ and $n$ which are the sum of two squares. Landau, and Ramanujan have proven independently, that $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$

And the constant of proportionality is the Landau-Ramanujan constant, which is a suspected transcendental.

Is there an exact formula for the function $\lambda(n)$? That is, is there an exact formula for the number of numbers between $0$ and $n$ which are the sum of two squares?

(On a lighter note, I wanted to ask another question which has been bothering me for quite a while. From what I have seen so far, (from the eyes of an absolute novice), in the analytic theory of numbers, I feel that there is more emphasis(or is there?)on finding out the asymptotic behavior of functions like $\lambda(n)$, rather than it's exact formula. Is this because of the difficulty of finding an exact formula? I must apologize if this question seems trivial. I just wanted some clarification and some advice on whether this was true.)

Thanks again in advance.

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    $\begingroup$ You're right, there is no exact formula known; this problem seems a lot harder than just finding the asymptotic. $\endgroup$ – David Hansen Apr 27 '11 at 3:39
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    $\begingroup$ What does it mean to find an "exact formula" anyway? $\endgroup$ – Matt Young Apr 27 '11 at 4:17
  • $\begingroup$ Koundinya, you might do better borrowing some books for beginners on this topic. Perhaps you can find Hardy and Wright, the fifth edition, title "An Introduction to the Theory of Numbers." The internet (and this web site in particular) does not provide effective tutoring for students prior to university. Or, put another way, over time you must move from appreciating the work of others to learning how to work in mathematics yourself. $\endgroup$ – Will Jagy Apr 27 '11 at 4:33
  • $\begingroup$ @Matt Young: By an exact formula, I mean an analytic expression which gives the exact number of integers in a given interval which are the sum of two squares. @Will Jagy: Thank you for your advice sir. I was just curious if an exact formula existed, and since nowhere was it mentioned that finding such a formula was hard, I naturally assumed that it was less interesting than finding the function's asymptotic. $\endgroup$ – Koundinya Vajjha Apr 27 '11 at 5:09
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    $\begingroup$ Alright. Much harder...a complete proof of the asymptotic formula appears in volume 2 of "Topics in Number Theory" by William J. LeVeque, in chapter 7, section 5. The book is now available from Dover, two volumes combined into one paperback. For comparison, there is, in some optimistic sense, an "exact formula" for the number of primes up to a bound. The work that went into that leads to the Prime Number Theorem, an asymptotic result, in turn to the Riemann Hypothesis. See "Riemann's Zeta Function" by Edwards, section 1.17, and "The Distribution of Prime Numbers" by A. E. Ingham. $\endgroup$ – Will Jagy Apr 27 '11 at 5:50

To answer your "lighter note" question, of course we want to find exact formulas. It's only when we can't find an exact (and useful) formula that we settle for asymptotic formulas, and if we can't even get those, we settle for upper and/or lower bounds. We do the best we can.


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