Which quaternary quadratic form represents $n$ the greatest number of times?

Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies $$r_Q(n)\ll_{k,\epsilon}n^{k/2-1+\epsilon}.$$ The implied constant depends only on $k$ and $\epsilon$, so it is independent of the actual coefficients of $Q$.

Remark. The "Added 1" section, posted with the permission of Valentin Blomer, contains a more precise result for $k=4$.

Proof. We induct on $k$, and for simplicity we do not indicate the dependence of implied constants on $k$. The case $k=2$ is classical and goes back to Gauss (see the "Added 2" section for more details). So let $k\geq 3$, and assume that the statement holds with $k-1$ in place of $k$. We can assume that $$ Q(x_1,\dots,x_k)=\sum_{1\leq i,j\leq k} a_{ij}x_ix_j$$ is Minkowski reduced. In particular, $a_{ij}=a_{ji}$ and $$ 0<a_{11}\leq a_{22}\leq\dots\leq a_{kk}. $$ Then we have a decomposition $$ Q(x_1,\dots,x_k)=\sum_{i=1}^k h_i\left(\sum_{i\leq j\leq k}c_{ij}x_j\right)^2,$$ where $h_i\asymp a_{ii}$, $c_{ii}=1$ and $c_{ij}\ll 1$ (see Theorem 3.1 and Lemma 1.3 in Chapter 12 of Cassels: Rational quadratic forms). In particular, the coefficients of $Q$ satisfy $$ a_{11}\dots a_{kk}\asymp h_1\dots h_k=\det(Q),$$ hence also $a_{ij}\ll a_{kk}\ll\det(Q)$ and $h_k\asymp a_{kk}\gg\det(Q)^{1/k}$.

We fix the positive integer $n$, and we consider the integral representations $Q(x_1,\dots,x_k)=n$. The number of representations with $x_k=0$ is $\ll_{\epsilon}n^{k/2-3/2+\epsilon}$ by the induction hypothesis, so we can focus on the representations with $x_k\neq 0$. From the above, we see immediately that $x_k\ll\sqrt{n}\det(Q)^{-1/(2k)}$, and then also that $x_{k-1}\ll\sqrt{n}$, then $x_{k-2}\ll\sqrt{n}$, and so on, finally $x_3\ll\sqrt{n}$. It follows that there are $\ll n^{(k-2)/2}\det(Q)^{-1/(2k)}$ choices for the $(k-2)$-tuple $(x_3,\dots,x_k)$ such that $x_k\neq 0$. Fixing such a tuple, we are left with an inhomogeneous binary representation problem $$ a_{11}x_1^2 + 2a_{12}x_1x_2 + a_{22}x_2^2 + d_1 x_1 + d_2 x_2 + e = 0 $$ with fixed integral coefficients $d_1,d_2\ll\sqrt{n}\det(Q)$ and $e\ll n\det(Q)$. Using Lemma 8 in this paper of Blomer and Pohl, it follows that the number of choices for $(x_1,x_2)$ is $\ll_\epsilon n^\epsilon\det(Q)^\epsilon$. Summing up, we get $$ r_Q(n)\ll_{\epsilon} n^{k/2-3/2+\epsilon} + n^{(k-2)/2+\epsilon}\det(Q)^{-1/(2k)+\epsilon} \ll n^{k/2-1+\epsilon},$$ and we are done.

Added 1. I have been in touch with Valentin Blomer about the original question, and my answer above incorporated a key input from him. More importantly, he realized that the above argument together with some automorphic input allows one to prove, for the case of $k=4$ variables, the striking uniform upper bound (with an absolute implied constant) $$r_Q(n) \ll \sigma(n).$$ Here is the argument of Valentin Blomer, posted with his permission. For $n\leq\det(Q)^{10}$, the last line of the inductive proof above gives $$ r_Q(n)\ll_{\epsilon} n^{1/2+\epsilon} + n^{1+\epsilon}\det(Q)^{-1/8+\epsilon} \ll n^{79/80+2\epsilon},$$ so we can (and we shall) assume that $n>\det(Q)^{10}$. We decompose the $\theta$-series of $Q$ uniquely as $$\theta_Q(z) = E(z) +S(z) = \sum_{n=1}^\infty a(n) e(nz) + \sum_{n=1}^\infty b(n) e(nz)$$ into an Eisenstein series and a cusp form of weight $2$ and level $N$, which is the level of $Q$. Accordingly, $r_Q(n)=a(n)+b(n)$, so it suffices to show that $a(n)\ll\sigma(n)$ and $b(n)\ll\sigma(n)$. The first bound was proved by Gogishvili (Georgian Math. J. 13 (2006), 687-691.), as follows from (2) and (13)-(14) in his paper. Therefore, it suffices to prove the second bound. We write $$S = \sum_{f \in B} c(f) f$$ in terms of an orthonormal Hecke eigenbasis $B$ for $S_2(N, \chi)$, where $\chi$ is a quadratic character and the inner product is given by $$(f, g) = \int_{\Gamma_0(N)\backslash \mathcal{H}} f(z)\bar{g}(z) \frac {dx\, dy}{y^2}.$$ We write $f(z) = \sum_n \lambda_f(n) e(nz)$, so that $b(n) = \sum_f c(f) \lambda_f(n)$. We avoid any use of Eichler/Deligne, among other things because it would require us to deal with oldforms carefully. Instead, we use the Petersson formula and Weil's bound for Kloosterman sums (together with Cauchy-Schwarz and Parseval): $$\begin{split} |b(n) |^2 \| S \|_2^{-2} n^{-1} & \leq n^{-1} \sum_f |\lambda_f(n)|^2 \ll 1 + \sum_{c} \frac{1}{c} S_{\chi}(n, n, c) J_1\left(\frac{4\pi n}{c}\right)\\ & \ll 1 + \sum_{c} \frac{(n, c)^{1/2}\tau(c)}{c^{1/2}} \min\left(\frac{n}{c}, \frac{c^{1/2}}{n^{1/2}}\right) \ll_\epsilon n^{1/2 + \epsilon}, \end{split}$$ so that $$b(n) \ll_\epsilon \| S \|_2 n^{3/4 + \epsilon}.$$ We have, by Lemma 4.2 of Blomer (Acta Arith. 114 (2004), 1-21.), $$\| S \|_2 \ll_\epsilon \det(Q)^{2+\epsilon},$$ whence in the end $$b(n) \ll_\epsilon \det(Q)^{2+\epsilon} n^{3/4 + \epsilon} \leq n^{19/20+2\epsilon}.$$ This concludes the proof. We note that for the twisted Kloosterman sum, the Weil-Estermann bound is not always true for higher prime powers, see Section 9 of Knightly-Li (Mem. Amer. Math. Soc. 224 (2013), no. 1055), but it is true for the case of quadratic characters that we are using here.

Added 2. Let me provide the details for the case $k=2$. Without loss of generality, $$Q(x,y)=ax^2+bxy+cy^2$$ is a reduced form. That is, $$|b|\leq a\leq c,\qquad\text{whence also}\qquad a\ll\det(Q)^{1/2}.$$ The equation $Q(x,y)=n$ can be rewritten as $$(2ax+by)^2+4\det(Q)y^2=4an.$$ We can assume that there are (integral) solutions with nonzero $y$, for otherwise there are at most two solutions. In this case, $$n\geq\det(Q)/a\gg a.$$ The equation factors in the ring of integers of an imaginary quadratic number field, hence a standard divisor bound argument combined with the previous display yields that the number of solutions is $$\ll_\epsilon(an)^\epsilon\ll_\epsilon n^{2\epsilon}.$$


GH from MO gave in his answer a bound due to himself and Valentin Blomer of $O(\sigma(n))$. I thought it would be interesting to compute the $O$. Here I'm looking for an effective bound of the form $\leq C \Delta^{-\delta} \sigma(n) + o(\sigma(n))$ where the $C$ is explicit but the little $o$ need not be, so we only need be concerned with the Eisenstein term.

The formula in the cited paper is $$\frac{\pi^2 n}{\sqrt{\Delta/16}} \prod_p \chi(p)$$

Define $f_Q(p)$ to be the max over $n$ of $\frac{\chi(p)}{p^{v_p(\Delta)/2}} (\sum_{t=0}^{v_p(n)} p^{-t}) $

Then the main term

$$\frac{4\pi^2 n}{\sqrt{\Delta}} \prod_p \chi(p) \leq 4\pi^2 \sigma(n) \prod_p f_Q(p)$$

So our goal is to upper bound $f_Q(p)$. According to the paper, for primes not dividing the discriminant $\Delta$, $\chi(p)= \left(1- \left(\frac{\Delta}{p} \right)p^{-2}\right) \sum_{t=0}^{v_p(n)}\left(\frac{\Delta}{p} \right)^t p^t$ so $f_Q(p) = 1- \left(\frac{\Delta}{p} \right)p^{-2}$. So the primes not dividing the discriminant contribute $\prod_p\left(1- \left(\frac{\Delta}{p} p^{-2} \right) \right)= L(\chi_{\sqrt{\Delta}},2)^{-1}$. However for a crude upper bound, we instead use $\chi(p) \leq 1+1/p^2$.

For odd primes dividing the discriminant, the bounds given for $\chi(p)$ depend on whether $p$ divides $n$ or not. If $p$ does not divide $n$, they depend further on $n_1$, which is the rank of the quadratic form mod $p$. In this case $\chi(p) \leq 2$ if the rank is $1$ but is at most $1+1/p$ otherwise. The rank can only be $1$ if $v_p(d) \geq 3$, because each term in the formula for the determinant $d$ of the symmetric matrix will be divisible by $p^3$. If $p$ does not divide $n$, the bound for $f_Q(p)$ is $p^{v_p(d)/6} (1+p^{-2}) (1+p^{-1})$. So we have

$$ f_Q(p) \leq \max \left( \frac{1 + 1/p}{p^{v_p(d)/2}}, \frac{1 + 1_{v_p(d) \geq 3}}{p^{v_p(d)/2}}, \frac{ 1 + p^{-2}}{p^{v_p(d)/3}}\right)$$

The third contribution is always the greatest as long as $(1+1/p) \leq p^{1/6} (1+ 1/p^2)$, which happens for all $p>2$, and $2 \leq \sqrt{p} (1+1/p^2)$, which happens for $p>3$ and only fails by a factor of $.962$ for $p=3$.

The remaining contribution is the local contribution at the prime $2$. The paper gives the bound $4 \cdot 2^{ (v_2(\Delta)-4)/6}$ for $\chi(2)$ and hence $f_Q(2) \leq 2^{4/3} 2^{-v_2(\Delta)/3}$.

Hence $$\prod_p f_Q(p) \leq \frac{2^{4/3}}{(1+1/4)}\frac{2}{\sqrt{3} (1+ 1/9)}\frac{1}{\Delta^{1/3}} \prod_p (1+1/p^2)$$

Using

$$\prod_p (1+ p^{-2}) = \frac{\zeta(2)}{\zeta(4)}=\frac{15}{\pi^2}$$ we get an upper bound for the main term of

$$ \frac{2^{4/3}}{(1+1/4)}\frac{2}{\sqrt{3} (1+ 1/9)}\frac{60 \sigma(n)}{\Delta^{1/3}}= 125.697\dots \frac{\sigma(n)}{\Delta^{1/3}}$$

To get the ratio under $30$, then, we need $\Delta>73$.

Combined with Valentin and GH's arguments, I believe this implies that there are only finitely many counterexamples to "the number of representatives is at most $30 \sigma(n)$" with $\Delta>73$.

It might be possible to prove a sharp bound by:

1) bounding the error term explicitly to eliminate large $\Delta$ counterexamples.

2) Explicitly calculating the main term for medium $\Delta$, instead of using this crude bound, to eliminate medium $\Delta$ counterexamples.

3) Explicitly calculating the main term and showing the error term vanishes for small $\Delta$, explicitly calculating the highest examples.


In a comment under Will Sawin's answer, GH says that it should not be too difficult to find a (relatively small) constant $C$ and a proof for $$ r_Q(n) \leq \, C \; \sigma(n) \, \det(Q)^{-1/9} \; + \; n^{4/5} $$ which, if $C$ were found to be small enough, would give teeth to the computations below.

I did want to see the behavior of specific forms of low discriminant from Nipp's tables, as Jeremy briefly indicated in an email. To get $r(n) \geq 15 \sigma(n)$ we seem to need discriminant $d \leq 21.$ To get $r(n) \geq 20 \sigma(n)$ we seem to need discriminant $d = 4,5.$

I should add that there are infinitely many forms that give $r(1) = 12,$ so that this ratio is at least $12.$ Given any positive integer $T \geq 2,$ $$ ( x^2 + y^2 + z^2 + yz + zx + xy) + T w^2 $$ represents $1$ twelve times.

Discriminant $4$ achieves the ratio $24.$ For $d=5,$ with prime $p \equiv \pm 2 \pmod 5,$ we get $$ r(p) = 30 (p-1) = 30 \; \sigma(p) \cdot \left( \frac{p-1}{p+1} \right) $$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

d = 5
                                          n      reps     sigma 
 ratio  20                                1       20        1  1 =  1 
 ratio  10                                2       30        3  2 = 2
 ratio  15                                3       60        4  3 = 3
 ratio  20                                5      120        6  5 = 5
 ratio  22.5                              7      180        8  7 = 7
 ratio  25.7142857142857                 13      360       14  13 = 13
 ratio  26.6666666666667                 17      480       18  17 = 17
 ratio  27.5                             23      660       24  23 = 23
 ratio  28.4210526315789                 37     1080       38  37 = 37
 ratio  28.6363636363636                 43     1260       44  43 = 43
 ratio  28.75                            47     1380       48  47 = 47
 ratio  28.8888888888889                 53     1560       54  53 = 53
 ratio  29.1176470588235                 67     1980       68  67 = 67
 ratio  29.1891891891892                 73     2160       74  73 = 73
 ratio  29.2857142857143                 83     2460       84  83 = 83
 ratio  29.3877551020408                 97     2880       98  97 = 97
 ratio  29.4230769230769                103     3060      104  103 = 103
 ratio  29.4444444444444                107     3180      108  107 = 107
 ratio  29.4736842105263                113     3360      114  113 = 113
 ratio  29.53125                        127     3780      128  127 = 127

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  d  4 record ratio  24 number  1       sigma  1   reps  24
  d  5 record ratio  29.6969696969697 number  197       sigma  198   reps  5880
  d  8 record ratio  17.8181818181818 number  197       sigma  198   reps  3528
  d  9 record ratio  12 number  1       sigma  1   reps  12
  d  12 record ratio  19.4117647058824 number  67       sigma  68   reps  1320
  d  12 record ratio  19.7979797979798 number  197       sigma  198   reps  3920
  d  13 record ratio  13.8585858585859 number  197       sigma  198   reps  2744
  d  16 record ratio  8 number  1       sigma  1   reps  8
  d  16 record ratio  12 number  1       sigma  1   reps  12
  d  17 record ratio  8.91 number  199       sigma  200   reps  1782
  d  20 record ratio  17.8181818181818 number  197       sigma  198   reps  3528
  d  20 record ratio  13.7142857142857 number  194       sigma  294   reps  4032
  d  20 record ratio  11.8787878787879 number  197       sigma  198   reps  2352
  d  21 record ratio  15.8383838383838 number  197       sigma  198   reps  3136
  d  21 record ratio  15.84 number  199       sigma  200   reps  3168
  d  24 record ratio  11.88 number  199       sigma  200   reps  2376
  d  24 record ratio  12 number  1       sigma  1   reps  12
  d  24 record ratio  11.8666666666667 number  179       sigma  180   reps  2136
  d  25 record ratio  6 number  1       sigma  1   reps  6
  d  28 record ratio  9.89010989010989 number  181       sigma  182   reps  1800
  d  28 record ratio  12 number  1       sigma  1   reps  12
  d  28 record ratio  9.89583333333333 number  191       sigma  192   reps  1900
  d  29 record ratio  10 number  5       sigma  6   reps  60
  d  29 record ratio  12 number  1       sigma  1   reps  12
  d  32 record ratio  9.8989898989899 number  197       sigma  198   reps  1960
  d  32 record ratio  11.8787878787879 number  197       sigma  198   reps  2352
  d  32 record ratio  12 number  1       sigma  1   reps  12
  d  32 record ratio  8.46031746031746 number  166       sigma  252   reps  2132
  d  32 record ratio  11.8666666666667 number  179       sigma  180   reps  2136
  d  33 record ratio  8 number  1       sigma  1   reps  8
  d  33 record ratio  7.91752577319588 number  193       sigma  194   reps  1536
  d  33 record ratio  7.91666666666667 number  191       sigma  192   reps  1520
  d  36 record ratio  12 number  1       sigma  1   reps  12
  d  36 record ratio  12 number  5       sigma  6   reps  72
  d  36 record ratio  8 number  1       sigma  1   reps  8
  d  36 record ratio  11.9504132231405 number  81       sigma  121   reps  1446
  d  36 record ratio  7.9843137254902 number  128       sigma  255   reps  2036
  d  37 record ratio  12 number  1       sigma  1   reps  12
  d  37 record ratio  7.54166666666667 number  191       sigma  192   reps  1448
  d  40 record ratio  12 number  1       sigma  1   reps  12
  d  40 record ratio  7.75257731958763 number  193       sigma  194   reps  1504
  d  40 record ratio  7.92156862745098 number  101       sigma  102   reps  808
  d  40 record ratio  7.84 number  149       sigma  150   reps  1176
  d  41 record ratio  8 number  1       sigma  1   reps  8
  d  41 record ratio  5.33333333333333 number  2       sigma  3   reps  16
  d  44 record ratio  8.64 number  149       sigma  150   reps  1296
  d  44 record ratio  8.55172413793103 number  173       sigma  174   reps  1488
  d  44 record ratio  12 number  1       sigma  1   reps  12
  d  44 record ratio  8.53658536585366 number  163       sigma  164   reps  1400
  d  45 record ratio  11.8787878787879 number  197       sigma  198   reps  2352
  d  45 record ratio  12 number  1       sigma  1   reps  12
  d  45 record ratio  11.8762886597938 number  193       sigma  194   reps  2304
  d  45 record ratio  11.5555555555556 number  159       sigma  216   reps  2496
  d  45 record ratio  11.8787878787879 number  197       sigma  198   reps  2352
  d  48 record ratio  11.8787878787879 number  197       sigma  198   reps  2352
  d  48 record ratio  9.8989898989899 number  197       sigma  198   reps  1960
  d  48 record ratio  9.8989898989899 number  197       sigma  198   reps  1960
  d  48 record ratio  12 number  1       sigma  1   reps  12
  d  48 record ratio  9.9 number  199       sigma  200   reps  1980
  d  48 record ratio  9.12592592592593 number  178       sigma  270   reps  2464
  d  48 record ratio  11.88 number  199       sigma  200   reps  2376
  d  48 record ratio  12 number  11       sigma  12   reps  144
  d  48 record ratio  9.1 number  158       sigma  240   reps  2184
  d  49 record ratio  4 number  1       sigma  1   reps  4
  d  52 record ratio  6.53061224489796 number  194       sigma  294   reps  1920
  d  52 record ratio  6.40740740740741 number  142       sigma  216   reps  1384
  d  52 record ratio  12 number  1       sigma  1   reps  12
  d  52 record ratio  8.36842105263158 number  151       sigma  152   reps  1272
  d  52 record ratio  5.57894736842105 number  151       sigma  152   reps  848
  d  52 record ratio  6 number  1       sigma  1   reps  6
  d  53 record ratio  8.12903225806452 number  61       sigma  62   reps  504
  d  53 record ratio  12 number  1       sigma  1   reps  12
  d  53 record ratio  8 number  11       sigma  12   reps  96
  d  56 record ratio  8 number  1       sigma  1   reps  8
  d  56 record ratio  7.31578947368421 number  151       sigma  152   reps  1112
  d  56 record ratio  7.46666666666667 number  89       sigma  90   reps  672
  d  56 record ratio  12 number  1       sigma  1   reps  12
  d  56 record ratio  7.15151515151515 number  131       sigma  132   reps  944
  d  57 record ratio  8 number  1       sigma  1   reps  8
  d  57 record ratio  5.67708333333333 number  191       sigma  192   reps  1090
  d  57 record ratio  5.69072164948454 number  193       sigma  194   reps  1104
  d  57 record ratio  6 number  1       sigma  1   reps  6
  d  60 record ratio  9.89690721649485 number  193       sigma  194   reps  1920
  d  60 record ratio  12 number  1       sigma  1   reps  12
  d  60 record ratio  9.89690721649485 number  193       sigma  194   reps  1920
  d  60 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  60 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  60 record ratio  9.9 number  199       sigma  200   reps  1980
  d  60 record ratio  9.9 number  199       sigma  200   reps  1980
  d  60 record ratio  9.88095238095238 number  167       sigma  168   reps  1660
  d  61 record ratio  12 number  1       sigma  1   reps  12
  d  61 record ratio  6 number  5       sigma  6   reps  36
  d  61 record ratio  6 number  1       sigma  1   reps  6
  d  64 record ratio  6 number  1       sigma  1   reps  6
  d  64 record ratio  12 number  1       sigma  1   reps  12
  d  64 record ratio  6.66666666666667 number  5       sigma  6   reps  40
  d  64 record ratio  4 number  1       sigma  1   reps  4
  d  64 record ratio  6 number  1       sigma  1   reps  6
  d  64 record ratio  6 number  1       sigma  1   reps  6
  d  64 record ratio  6 number  3       sigma  4   reps  24
  d  65 record ratio  8 number  1       sigma  1   reps  8
  d  65 record ratio  5.24137931034483 number  173       sigma  174   reps  912
  d  65 record ratio  5.28 number  149       sigma  150   reps  792
  d  65 record ratio  6 number  1       sigma  1   reps  6
  d  68 record ratio  9.6 number  19       sigma  20   reps  192
  d  68 record ratio  12 number  1       sigma  1   reps  12
  d  68 record ratio  8 number  1       sigma  1   reps  8
  d  68 record ratio  6.09523809523809 number  41       sigma  42   reps  256
  d  68 record ratio  6.12371134020619 number  193       sigma  194   reps  1188
  d  68 record ratio  5.33888888888889 number  184       sigma  360   reps  1922
  d  68 record ratio  5.32549019607843 number  128       sigma  255   reps  1358
  d  69 record ratio  8.57142857142857 number  13       sigma  14   reps  120
  d  69 record ratio  12 number  1       sigma  1   reps  12
  d  69 record ratio  7.92 number  199       sigma  200   reps  1584
  d  69 record ratio  7.92 number  199       sigma  200   reps  1584
  d  69 record ratio  7.91919191919192 number  197       sigma  198   reps  1568
  d  72 record ratio  8 number  1       sigma  1   reps  8
  d  72 record ratio  7.33333333333333 number  83       sigma  84   reps  616
  d  72 record ratio  12 number  1       sigma  1   reps  12
  d  72 record ratio  7.16483516483517 number  181       sigma  182   reps  1304
  d  72 record ratio  7.13924050632911 number  157       sigma  158   reps  1128
  d  72 record ratio  7.33333333333333 number  83       sigma  84   reps  616
  d  72 record ratio  7.15151515151515 number  131       sigma  132   reps  944
  d  72 record ratio  7 number  159       sigma  216   reps  1512
  d  72 record ratio  6.96774193548387 number  183       sigma  248   reps  1728
  d  73 record ratio  4 number  1       sigma  1   reps  4
  d  73 record ratio  6 number  1       sigma  1   reps  6
  d  73 record ratio  4 number  2       sigma  3   reps  12
  d  76 record ratio  12 number  1       sigma  1   reps  12
  d  76 record ratio  5.42857142857143 number  139       sigma  140   reps  760
  d  76 record ratio  5.24390243902439 number  163       sigma  164   reps  860
  d  76 record ratio  6 number  1       sigma  1   reps  6
  d  76 record ratio  5.31868131868132 number  181       sigma  182   reps  968
  d  76 record ratio  6 number  1       sigma  1   reps  6
  d  77 record ratio  12 number  1       sigma  1   reps  12
  d  77 record ratio  8.10989010989011 number  181       sigma  182   reps  1476
  d  77 record ratio  7.96153846153846 number  103       sigma  104   reps  828
  d  77 record ratio  8.18181818181818 number  109       sigma  110   reps  900
  d  77 record ratio  8.08 number  149       sigma  150   reps  1212
  d  80 record ratio  12 number  1       sigma  1   reps  12
  d  80 record ratio  9 number  167       sigma  168   reps  1512
  d  80 record ratio  6 number  1       sigma  1   reps  6
  d  80 record ratio  8 number  1       sigma  1   reps  8
  d  80 record ratio  7.4639175257732 number  193       sigma  194   reps  1448
  d  80 record ratio  8.90909090909091 number  197       sigma  198   reps  1764
  d  80 record ratio  5.93939393939394 number  197       sigma  198   reps  1176
  d  80 record ratio  7.58823529411765 number  67       sigma  68   reps  516
  d  80 record ratio  7.42307692307692 number  103       sigma  104   reps  772
  d  80 record ratio  5.87755102040816 number  194       sigma  294   reps  1728
  d  80 record ratio  5.87755102040816 number  194       sigma  294   reps  1728
  d  80 record ratio  7.39285714285714 number  188       sigma  336   reps  2484
  d  81 record ratio  8 number  1       sigma  1   reps  8
  d  81 record ratio  4.7 number  19       sigma  20   reps  94
  d  81 record ratio  4 number  1       sigma  1   reps  4
  d  81 record ratio  6 number  1       sigma  1   reps  6
  d  81 record ratio  6 number  2       sigma  3   reps  18
  d  84 record ratio  12 number  1       sigma  1   reps  12
  d  84 record ratio  9.52747252747253 number  181       sigma  182   reps  1734
  d  84 record ratio  7.35135135135135 number  146       sigma  222   reps  1632
  d  84 record ratio  7.30612244897959 number  194       sigma  294   reps  2148
  d  84 record ratio  8 number  1       sigma  1   reps  8
  d  84 record ratio  6.4 number  179       sigma  180   reps  1152
  d  84 record ratio  6.33333333333333 number  191       sigma  192   reps  1216
  d  84 record ratio  6.35164835164835 number  181       sigma  182   reps  1156
  d  84 record ratio  6.36 number  199       sigma  200   reps  1272
  d  84 record ratio  7.25925925925926 number  142       sigma  216   reps  1568
  d  84 record ratio  7.25925925925926 number  142       sigma  216   reps  1568
  d  84 record ratio  9.56521739130435 number  137       sigma  138   reps  1320
  d  84 record ratio  9.6 number  179       sigma  180   reps  1728
  d  85 record ratio  12 number  1       sigma  1   reps  12
  d  85 record ratio  6.26086956521739 number  137       sigma  138   reps  864
  d  85 record ratio  6.03846153846154 number  103       sigma  104   reps  628
  d  85 record ratio  6.10909090909091 number  109       sigma  110   reps  672
  d  85 record ratio  6.4 number  29       sigma  30   reps  192
  d  85 record ratio  6.10989010989011 number  181       sigma  182   reps  1112
  d  88 record ratio  12 number  1       sigma  1   reps  12
  d  88 record ratio  4.69230769230769 number  103       sigma  104   reps  488
  d  88 record ratio  6 number  1       sigma  1   reps  6
  d  88 record ratio  4.68 number  199       sigma  200   reps  936
  d  88 record ratio  6 number  1       sigma  1   reps  6
  d  88 record ratio  6 number  1       sigma  1   reps  6
  d  88 record ratio  5 number  3       sigma  4   reps  20
  d  89 record ratio  8 number  1       sigma  1   reps  8
  d  89 record ratio  4.22222222222222 number  17       sigma  18   reps  76
  d  89 record ratio  4 number  1       sigma  1   reps  4
  d  89 record ratio  3.71428571428571 number  4       sigma  7   reps  26
  d  92 record ratio  12 number  1       sigma  1   reps  12
  d  92 record ratio  6 number  1       sigma  1   reps  6
  d  92 record ratio  8 number  1       sigma  1   reps  8
  d  92 record ratio  6.46666666666667 number  29       sigma  30   reps  194
  d  92 record ratio  5.93406593406593 number  181       sigma  182   reps  1080
  d  92 record ratio  6 number  1       sigma  1   reps  6
  d  92 record ratio  5.93333333333333 number  179       sigma  180   reps  1068
  d  93 record ratio  12 number  1       sigma  1   reps  12
  d  93 record ratio  7.15714285714286 number  139       sigma  140   reps  1002
  d  93 record ratio  7.125 number  31       sigma  32   reps  228
  d  93 record ratio  8 number  1       sigma  1   reps  8
  d  93 record ratio  7.5 number  23       sigma  24   reps  180
  d  93 record ratio  7.02777777777778 number  71       sigma  72   reps  506
  d  96 record ratio  12 number  1       sigma  1   reps  12
  d  96 record ratio  8.45454545454546 number  43       sigma  44   reps  372
  d  96 record ratio  6.88888888888889 number  107       sigma  108   reps  744
  d  96 record ratio  6.72463768115942 number  137       sigma  138   reps  928
  d  96 record ratio  8 number  1       sigma  1   reps  8
  d  96 record ratio  7.91208791208791 number  181       sigma  182   reps  1440
  d  96 record ratio  7.91208791208791 number  181       sigma  182   reps  1440
  d  96 record ratio  6.6 number  199       sigma  200   reps  1320
  d  96 record ratio  7.92 number  199       sigma  200   reps  1584
  d  96 record ratio  7.92 number  199       sigma  200   reps  1584
  d  96 record ratio  7.92 number  199       sigma  200   reps  1584
  d  96 record ratio  5.73333333333333 number  118       sigma  180   reps  1032
  d  96 record ratio  5.65079365079365 number  166       sigma  252   reps  1424
  d  96 record ratio  5.63333333333333 number  158       sigma  240   reps  1352
  d  96 record ratio  5.63333333333333 number  158       sigma  240   reps  1352
  d  96 record ratio  7.88405797101449 number  137       sigma  138   reps  1088
  d  96 record ratio  8 number  5       sigma  6   reps  48
  d  96 record ratio  7.91111111111111 number  179       sigma  180   reps  1424
  d  97 record ratio  4 number  1       sigma  1   reps  4
  d  97 record ratio  6 number  1       sigma  1   reps  6
  d  97 record ratio  3.5 number  3       sigma  4   reps  14
  d  97 record ratio  3.33333333333333 number  2       sigma  3   reps  10
  d  100 record ratio  12 number  1       sigma  1   reps  12
  d  100 record ratio  6 number  3       sigma  4   reps  24
  d  100 record ratio  6.28571428571429 number  13       sigma  14   reps  88
  d  100 record ratio  6 number  1       sigma  1   reps  6
  d  100 record ratio  7.97435897435897 number  125       sigma  156   reps  1244
  d  100 record ratio  6 number  1       sigma  1   reps  6
  d  100 record ratio  5 number  3       sigma  4   reps  20
  d  100 record ratio  3.9921568627451 number  128       sigma  255   reps  1018
  d  101 record ratio  12 number  1       sigma  1   reps  12
  d  101 record ratio  8 number  1       sigma  1   reps  8
  d  101 record ratio  6 number  19       sigma  20   reps  120
  d  101 record ratio  6 number  5       sigma  6   reps  36
  d  101 record ratio  6 number  1       sigma  1   reps  6
  d  104 record ratio  12 number  1       sigma  1   reps  12
  d  104 record ratio  5.06122448979592 number  97       sigma  98   reps  496
  d  104 record ratio  5.15463917525773 number  193       sigma  194   reps  1000
  d  104 record ratio  5.22448979591837 number  97       sigma  98   reps  512
  d  104 record ratio  6 number  1       sigma  1   reps  6
  d  104 record ratio  8 number  1       sigma  1   reps  8
  d  104 record ratio  5.33333333333333 number  5       sigma  6   reps  32
  d  104 record ratio  5.11392405063291 number  157       sigma  158   reps  808
  d  105 record ratio  8 number  1       sigma  1   reps  8
  d  105 record ratio  6 number  1       sigma  1   reps  6
  d  105 record ratio  5.27835051546392 number  193       sigma  194   reps  1024
  d  105 record ratio  5.37931034482759 number  173       sigma  174   reps  936
  d  105 record ratio  5.30952380952381 number  167       sigma  168   reps  892
  d  105 record ratio  5.375 number  191       sigma  192   reps  1032
  d  105 record ratio  5.26666666666667 number  179       sigma  180   reps  948
  d  105 record ratio  6 number  1       sigma  1   reps  6
  d  105 record ratio  5.28 number  199       sigma  200   reps  1056
  d  105 record ratio  6 number  1       sigma  1   reps  6
  d  105 record ratio  5.29032258064516 number  61       sigma  62   reps  328
  d  108 record ratio  12 number  1       sigma  1   reps  12
  d  108 record ratio  6.6 number  199       sigma  200   reps  1320
  d  108 record ratio  6.609375 number  127       sigma  128   reps  846
  d  108 record ratio  6.65853658536585 number  163       sigma  164   reps  1092
  d  108 record ratio  8 number  1       sigma  1   reps  8
  d  108 record ratio  7.46666666666667 number  29       sigma  30   reps  224
  d  108 record ratio  7.15909090909091 number  129       sigma  176   reps  1260
  d  108 record ratio  7.15909090909091 number  129       sigma  176   reps  1260
  d  108 record ratio  6.74747474747475 number  197       sigma  198   reps  1336
  d  108 record ratio  6.64367816091954 number  173       sigma  174   reps  1156
  d  108 record ratio  6.63636363636364 number  197       sigma  198   reps  1314
  d  108 record ratio  6.66666666666667 number  11       sigma  12   reps  80
  d  108 record ratio  9.9 number  199       sigma  200   reps  1980
  d  108 record ratio  7.22222222222222 number  159       sigma  216   reps  1560
  d  108 record ratio  9.8989898989899 number  197       sigma  198   reps  1960
  d  109 record ratio  12 number  1       sigma  1   reps  12
  d  109 record ratio  6 number  1       sigma  1   reps  6
  d  109 record ratio  5.33333333333333 number  5       sigma  6   reps  32
  d  109 record ratio  6 number  1       sigma  1   reps  6
  d  109 record ratio  4.5 number  3       sigma  4   reps  18
  d  112 record ratio  12 number  1       sigma  1   reps  12
  d  112 record ratio  6 number  3       sigma  4   reps  24
  d  112 record ratio  6 number  1       sigma  1   reps  6
  d  112 record ratio  5.93406593406593 number  181       sigma  182   reps  1080
  d  112 record ratio  6 number  1       sigma  1   reps  6
  d  112 record ratio  6 number  1       sigma  1   reps  6
  d  112 record ratio  5.02083333333333 number  191       sigma  192   reps  964
  d  112 record ratio  5.05494505494505 number  181       sigma  182   reps  920
  d  112 record ratio  5.05494505494505 number  181       sigma  182   reps  920
  d  112 record ratio  6 number  1       sigma  1   reps  6
  d  112 record ratio  5.06122448979592 number  97       sigma  98   reps  496
  d  112 record ratio  4.57142857142857 number  194       sigma  294   reps  1344
  d  112 record ratio  4.57142857142857 number  194       sigma  294   reps  1344
  d  112 record ratio  4.66666666666667 number  2       sigma  3   reps  14
  d  112 record ratio  5.9375 number  191       sigma  192   reps  1140
  d  112 record ratio  4.55 number  158       sigma  240   reps  1092
  d  113 record ratio  8 number  1       sigma  1   reps  8
  d  113 record ratio  4 number  1       sigma  1   reps  4
  d  113 record ratio  6 number  1       sigma  1   reps  6
  d  113 record ratio  3.71428571428571 number  4       sigma  7   reps  26
  d  113 record ratio  3.41666666666667 number  71       sigma  72   reps  246
  d  116 record ratio  8 number  1       sigma  1   reps  8
  d  116 record ratio  4.09523809523809 number  41       sigma  42   reps  172
  d  116 record ratio  4.66666666666667 number  5       sigma  6   reps  28
  d  116 record ratio  6 number  1       sigma  1   reps  6
  d  116 record ratio  4 number  7       sigma  8   reps  32
  d  116 record ratio  12 number  1       sigma  1   reps  12
  d  116 record ratio  6.66666666666667 number  5       sigma  6   reps  40
  d  116 record ratio  6.14285714285714 number  41       sigma  42   reps  258
  d  116 record ratio  6 number  1       sigma  1   reps  6
  d  116 record ratio  4.60215053763441 number  122       sigma  186   reps  856
  d  116 record ratio  4.66666666666667 number  2       sigma  3   reps  14
  d  117 record ratio  8 number  1       sigma  1   reps  8
  d  117 record ratio  7.33333333333333 number  17       sigma  18   reps  132
  d  117 record ratio  12 number  1       sigma  1   reps  12
  d  117 record ratio  6.92783505154639 number  193       sigma  194   reps  1344
  d  117 record ratio  6.94736842105263 number  151       sigma  152   reps  1056
  d  117 record ratio  6.92783505154639 number  193       sigma  194   reps  1344
  d  117 record ratio  6.04615384615385 number  171       sigma  260   reps  1572
  d  117 record ratio  6.06153846153846 number  171       sigma  260   reps  1576
  d  117 record ratio  6.97619047619048 number  167       sigma  168   reps  1172
  d  117 record ratio  6.98550724637681 number  137       sigma  138   reps  964
  d  120 record ratio  8 number  1       sigma  1   reps  8
  d  120 record ratio  6.25287356321839 number  173       sigma  174   reps  1088
  d  120 record ratio  6.30952380952381 number  167       sigma  168   reps  1060
  d  120 record ratio  12 number  1       sigma  1   reps  12
  d  120 record ratio  6.43298969072165 number  193       sigma  194   reps  1248
  d  120 record ratio  6.30927835051546 number  193       sigma  194   reps  1224
  d  120 record ratio  6.36734693877551 number  97       sigma  98   reps  624
  d  120 record ratio  6.48484848484848 number  131       sigma  132   reps  856
  d  120 record ratio  6.54545454545455 number  131       sigma  132   reps  864
  d  120 record ratio  6.4 number  89       sigma  90   reps  576
  d  120 record ratio  6.31111111111111 number  179       sigma  180   reps  1136
  d  120 record ratio  6.41758241758242 number  181       sigma  182   reps  1168
  d  120 record ratio  6.61818181818182 number  109       sigma  110   reps  728
  d  120 record ratio  6.46153846153846 number  181       sigma  182   reps  1176
  d  121 record ratio  4 number  1       sigma  1   reps  4
  d  121 record ratio  6 number  1       sigma  1   reps  6
  d  121 record ratio  4 number  2       sigma  3   reps  12
  d  124 record ratio  12 number  1       sigma  1   reps  12
  d  124 record ratio  6 number  1       sigma  1   reps  6
  d  124 record ratio  4.61904761904762 number  41       sigma  42   reps  194
  d  124 record ratio  4.19354838709677 number  25       sigma  31   reps  130
  d  124 record ratio  5.33333333333333 number  5       sigma  6   reps  32
  d  124 record ratio  4.30769230769231 number  9       sigma  13   reps  56
  d  124 record ratio  6 number  1       sigma  1   reps  6
  d  124 record ratio  4 number  47       sigma  48   reps  192
  d  124 record ratio  4.02439024390244 number  163       sigma  164   reps  660
  d  125 record ratio  8 number  1       sigma  1   reps  8
  d  125 record ratio  12 number  1       sigma  1   reps  12
  d  125 record ratio  6.15625 number  127       sigma  128   reps  788
  d  125 record ratio  6.32432432432432 number  73       sigma  74   reps  468
  d  125 record ratio  6 number  103       sigma  104   reps  624
  d  125 record ratio  6 number  1       sigma  1   reps  6
  d  125 record ratio  9.8989898989899 number  197       sigma  198   reps  1960
  d  125 record ratio  7.89473684210526 number  185       sigma  228   reps  1800

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