Comments in prime_nth_count and BENCH_SEGCOUNT changes segment prime …

…count

TODO

@@ -119,6 +119,8 @@
 - polymul, polyadd, polydiv, polyneg, polyeval, polyorder, polygcd, polylcm, polyroots, ...
   A lot of our ops do these mod n, we could make ..mod versions of each.
 
+- poly_is_reducible
+
 - use word-based for-sieve for non-segment.
 
 - remove start/end partial word tests from inner loop in for-sieve.
@@ -128,9 +130,14 @@
 - compare wheel_t with primes separated and possibly cached.
 
 - urandomm with bigints could be faster.
-   8.2s  my $f=factorial(144); urandomm($f) for 1..5e5;
+   7.7s  my $f=factorial(144); urandomm($f) for 1..5e5;
+   6.2s  my $f=factorial(144); urandomm("$f") for 1..5e5;
+   4.8s  my $f="".factorial(144); urandomm($f) for 1..5e5;
    5.3s  use Math::GMP qw/:constant/; my $f=factorial(144); urandomm($f) for 1..5e5;
-   1.8s  my $f=Math::Prime::Util::GMP::factorial(144); Math::Prime::Util::GMP::urandomm($f) for 1..5e5;
+   1.7s  my $f=Math::Prime::Util::GMP::factorial(144); Math::Prime::Util::GMP::urandomm($f) for 1..5e5;
+  In the first case, we're calling ""->bstr->_str once for validation in MPU
+  and and once for use in MPU::GMP.
+  The last case is all strings with no read/write bigint objects anywhere.
 
 - Destroy csprng context on thread destruction.
 
@@ -158,3 +165,9 @@
 - update prime count lower/upper from https://arxiv.org/pdf/1703.08032.pdf
 
 - urandomr
+
+- circular primes ... just use repdigits after 1M?  https://oeis.org/A068652
+
+- forfactored and forsquarefree.  Consider setting $_ and @_.
+  Would formoebius make more sense?
+  We have ranged_moebius which is efficient, but not segmented so must be careful.

prime_nth_count.c

@@ -336,6 +336,10 @@ UV prime_count(UV lo, UV hi)
   if (lo > hi || hi < 2)
     return 0;
 
+#if defined(BENCH_SEGCOUNT)
+  return segment_prime_count(lo, hi);
+#endif
+
   /* We use table acceleration so this is preferable for small inputs */
   if (hi < 66000000)  return segment_prime_count(lo, hi);
 
@@ -399,6 +403,9 @@ UV prime_count_lower(UV n)
   fl1 = logl(n);
   fl2 = fl1 * fl1;
 
+  /* Axler 2014: https://arxiv.org/abs/1409.1780  (v7 2016), Cor 3.6
+   * show variations of this. */
+
   if (n <= 300000) { /* Quite accurate and avoids calling Li for speed. */
     a = (n < 70200) ? 947 : (n < 176000) ? 904 : 829;
     lower = fn / (fl1 - 1 - 1/fl1 - 2.85/fl2 - 13.15/(fl1*fl2) + a/(fl2*fl2));
@@ -463,6 +470,12 @@ UV prime_count_upper(UV n)
   fl1 = logl(n);
   fl2 = fl1 * fl1;
 
+  /* Axler 2014: https://arxiv.org/abs/1409.1780  (v7 2016), Cor 3.5
+   *
+   * upper = fn/(fl1-1.0L-1.0L/fl1-3.35L/fl2-12.65L/(fl2*fl1)-89.6L/(fl2*fl2));
+   * return (UV) floorl(upper);
+   */
+
   if (BITS_PER_WORD == 32 || fn <= 821800000.0) {  /* Dusart 2010, page 2 */
     for (i = 0; i < (int)NUPPER_THRESH; i++)
       if (n < _upper_thresh[i].thresh)
@@ -514,7 +527,7 @@ UV nth_prime_upper(UV n)
   /* Dusart 2010 page 2 */
   upper = fn * (flogn + flog2n - 1.0 + ((flog2n-2.00)/flogn));
   if        (n >= 46254381) {
-     /* Axler 2017 http//arxiv.org/pdf/1706.03651.pdf Corollary 1.2 */
+     /* Axler 2017 http://arxiv.org/pdf/1706.03651.pdf Corollary 1.2 */
     upper -= fn * ((flog2n*flog2n-6*flog2n+10.667)/(2*flogn*flogn));
   } else if (n >=  8009824) {
     /* Axler 2013 page viii Korollar G */