Added von Mangoldt function

collects/math/private/number-theory/number-theory.rkt

@@ -44,6 +44,7 @@
          totient
          moebius-mu
          divisor-sum
+         mangoldt-lambda
          )
 
 ;;;
@@ -322,6 +323,7 @@
 ; Factor a number n without prime factors below the prime p.
 (define (small-prime-factors-over n p) ; p prime
   (cond
+    [(<= p 0)        (raise-argument-error 'small-prime-factors-over "Natural" p)]
     [(< n p)         '()]
     [(= n p)         (list (list p 1))]
     [(prime? n)      (list (list n 1))]
@@ -689,6 +691,14 @@
                               ps es))
                 natural?))])))
 
+(: mangoldt-lambda : Z -> Real)                   
+(define (mangoldt-lambda n)
+  (cond 
+    [(<= n 0) (raise-argument-error 'mangoldt-lambda "Natural" n)]
+    [else (define am (prime-power n))
+          (cond
+            [(cons? am) (log (car am))]
+            [else 0])]))
 
 ; These tests are for un-exported functions.
 #;(begin

collects/math/scribblings/math-number-theory.scrbl

@@ -561,6 +561,19 @@ Note: The function @racket[prime-omega] is multiplicative.
                     (prime-omega (* 2 2 2 3 3 5))]
 }
 
+@margin-note{Wikipedia: @hyperlink["http://en.wikipedia.org/wiki/Von_Mangoldt_function"]{Von Mangoldt Function}}
+@defproc[(mangoldt-lambda [n Natural]) Real]{
+The von Mangoldt function. 
+If @racket[n=p^k] for a prime @racket[p] and an integer @racket[k>=1] then @racket[(log n)] is returned.                                             
+Otherwise 0 is returned.
+
+@interaction[#:eval untyped-eval         
+                    (mangoldt-lambda (* 3 3))
+                    (log 3)]
+}
+
+
+
 @; ----------------------------------------
 @section[#:tag "number-sequences"]{Number Sequences}