How to resolve the algorithm Mertens function step by step in the Forth programming language

Published on 12 May 2024 09:40 PM
#Forth

How to resolve the algorithm Mertens function step by step in the Forth programming language

Table of Contents

Problem Statement

The Mertens function M(x) is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. It is an extension of the Möbius function. Given the Möbius function μ(n), the Mertens function M(x) is the sum of the Möbius numbers from n == 1 through n == x.

This is not code golf.   The stackexchange link is provided as an algorithm reference, not as a guide.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Mertens function step by step in the Forth programming language

Source code in the forth programming language

: AMOUNT 1000 ;

variable mertens AMOUNT cells allot
: M 1- cells mertens + ; \ 1-indexed array 

: make-mertens
  1 1 M !
  2 begin dup AMOUNT <= while
    1 over M !
    2 begin over over >= while
      over over / M @ 
      2 pick M @ swap -
      2 pick M !  
    1+ repeat
    drop
  1+ repeat
  drop
;
  
: print-row
  begin dup while 
    swap dup M @ 3 .r 1+
    swap 1-
  repeat
  drop
;

: print-table ."    "
  1 9 print-row cr
  begin dup 100 < while 10 print-row cr repeat
  drop
;

: find-zero-cross
  0 0
  1 begin dup AMOUNT <= while
    dup M @ 0= if
      swap 1+ swap
      dup 1- M @ 0<> if rot 1+ -rot then
    then
    1+
  repeat
  drop
;

make-mertens
." The first 99 Mertens numbers are:" cr print-table
find-zero-cross
." M(N) is zero " . ." times." cr
." M(N) crosses zero " . ." times." cr
bye

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