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Mertens function
Anant Saxena
December 28, 2015
1 Basic Definitions(n, r) = Remainder of
n
r
We start with a discrete variable n (n can only be an integer).
Hence,
n= 1 + 1 +. . . ntimes
Now, using the mobius inversion formula:
1 =n
r=1
(r)n
r
Where,x is the floor function on xWe will call this series basic
sum.
2 Mertens function
2.1 Series
We start by considering:
1
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n= 1 + 1 +. . . ntimes
n= (0 + 1) + . . . 2ntimes
n= (0 + 0 + 1) + . . . 3ntimes
...
n= (0 + 0 + 0 n1times
+1) + . . . n2times
(1)
Multiplying the rth equation with (r)
2.2 Example n =3
When n= 3
n= 1 + 1 + 1
n= 0 + 1 + 0 + 1 + 0 + 1
n= 0 + 0 + 1 + 0 + 0 + 1 + 0 + 0 + 1
(2)
Now we apply the following algorithm: we add another 0
appropriatelyto make it look as the the 1st nterms are repeating in
each equation :
n= 1 + 1 + 1
n= 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1(3,2)
n= 0 + 0 + 1 + 0 + 0 + 1 + 0 + 0 + 1
(3)
Now we add and 1s in each equation to make each equations
initial termsto look like the first n terms
n= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1(n1) n
1
n= 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0(n2)(n1)n
2
+ 1
(3,2)
n= 0 + 0 + 1 + 0 + 0 + 1 + 0 + 0 + 1
(4)
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Multiplying the rth equation with (r)
(1)n= (1)(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1(n1)n
1
)
(2)n= (2)(0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0(n2)n
2
+ 1
(3,2)
)
(3)n= (3)(0 + 0 + 1 + 0 + 0 + 1 + 0 + 0 + 1)
(5)
Adding all the above:
(1)n= (1)(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1(n1) n
1)(2)n= (2)(0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0(n2)
n
2
+ 1
(3,2)
)
(3)n= (3)(0 + 0 + 1 + 0 + 0 + 1 + 0 + 0+)(n n)n
3
)
(6)
We notice the first n= 3 terms is the basic sum. Hence,
(1)n= (1)(
1 + 1 + 1 +
1 + 1 + 1 +
1 + 1 + 1 (n1)n
1
)
(2)n= (2)(0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0(n2) n2+ 1(3,2)
)
(3)n= (3)(0 + 0 + 1 =1
+ 0 + 0 + 1 =1
+ 0 + 0 + 1) =1
(n n)n
3
)
(7)
2.3 Generalization of Sum
Adding the a generalization of the above equations together:
n
n
r=1
(r) = nn
r=1
(r)(n r) n
r +n
r=1
(r)(n, r)
= nn
r=1
(r) =n
r=1
(r)rn
r
+
nr=1
(r)(n, r)
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= nn
r=1
(r) =
n
r=1
(r)r nr+
n
r=1
(r)(n, r)
= nn
r=1
(r) =n
r=1
(r)rn
r
+
nr=1
(r)(nn
r
r)
IDEA FAIL :(
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