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The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper discusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

Added. One can turn Greg Martin's response into a rigorous disproof of the added conjecture as follows. Assume that the conjecture is true, then in any interval $(a+k\sqrt{a},a+(k+1)\sqrt{a})$ with $0\leq k\leq\sqrt{a}$ the number of $n$'s with $r(n)>0$ is $\Omega(\sqrt{a})$. This implies that in $(a,2a)$ the number of $n$'s with $r(n)>0$ is $\Omega(a)$, contradicting Landau's result $O(a/\sqrt{\log a})$ for the number of such $n$'s.

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper discusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper discusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

Added. One can turn Greg Martin's response into a rigorous disproof of the added conjecture as follows. Assume that the conjecture is true, then in any interval $(a+k\sqrt{a},a+(k+1)\sqrt{a})$ with $0\leq k\leq\sqrt{a}$ the number of $n$'s with $r(n)>0$ is $\Omega(\sqrt{a})$. This implies that in $(a,2a)$ the number of $n$'s with $r(n)>0$ is $\Omega(a)$, contradicting Landau's result $O(a/\sqrt{\log a})$ for the number of such $n$'s.

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GH from MO
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper disussesdiscusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper disusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper discusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.

Source Link
GH from MO
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

The topic is too general, and your question is too vague.

At any rate, for a large $x$, the average value of $r(n)$ for $1\leq n\leq x$ is about $\pi$, while the standard deviation is about $2\sqrt{\log x}$. There are many ways to prove this, perhaps the most instructive is to look at the Dirichlet series $\sum r(n)n^{-s}$ and $\sum r(n)^2n^{-s}$ which have poles of order $1$ and $2$, respectively (as follows from their Euler product decomposition). The following recent paper disusses the analogous problem in short intervals: Garaev-Kühleitner-Luca-Nowak, Asymptotic formulas for certain arithmetic functions, Math. Slovaca 58 (2008), 301–308.