C O L L O Q U I U M M A T H E M A T I C U M
VOL. LXIX 1995 FASC. 1
ON EULER–VON MANGOLDT’S EQUATION
BY
MARIUSZ S K A L B A (WARSZAWA)
The title equation
∞
X
n=1
µ(n) n = 0,
elementarily equivalent to the prime number theorem, was conjectured by Euler and proved by von Mangoldt. We shall prove here a much more general theorem:
Theorem. Let F be an algebraic number field and A an arbitrary non- empty set of prime ideals of the ring of integers O F . Then
(1) X
a
µ A (a) N (a) = Y
p∈A
1 − 1 N (p)
,
where the sum is extended over all integral ideals of O F and µ A is defined by µ A (a) = µ(a) if a is not divisible by any prime ideal outside A and µ A (a) = 0 otherwise.
Lemma 1 (Axer’s theorem, [1]). Let f (a) be an ideal-theoretic function satisfying
X
N (a)≤x
|f (a)| = O(x), X
N (a)≤x
f (a) = o(x).
Then
X
N (a)≤x
α x
N (a) −
x N (a)
F
f (a) = o(x), where
[x] F = |{a C O F : N (a) ≤ x}| = αx + O(x 1−1/k ), k = (F : Q), and α = α F .
1991 Mathematics Subject Classification: 11M41, 11R44.
[143]
144 M. S K A L B A
Lemma 2 (the simplest sieve). Let B be a set of prime ideals of O F such that
X
p∈B
1
N (p) = ∞.
Then
N P \B (x) = |{a C O F : N (a) ≤ x and p|a ⇒ p ∈ P \ B}| = o(x).
P r o o f o f t h e T h e o r e m. We shall distinguish three cases.
C a s e 1: P
p∈A 1/N (p) < ∞. This implies that the product on the right-hand side of (1) is absolutely convergent. Hence (1) holds.
C a s e 2: P
p∈P \A 1/N (p) < ∞. This case is a classical one; we multiply two convergent Dirichlet series, the second of them is absolutely convergent:
X
a
µ(a)
N (a) · X
a∈N
P \A1
N (a) = X
a
µ A (a) N (a) .
The first one is convergent to 0 by Landau [3] (the special case F = Q by von Mangoldt [4]) and because the above multiplication is allowed we obtain (1) also in this case.
C a s e 3: P
p∈A 1/N (p) = ∞ and P
p∈P \A 1/N (p) = ∞. This is the most interesting case. We estimate as follows:
αx X
N (a)≤x
µ A (a) N (a)
≤
X
N (a)≤x
α x
N (a) −
x N (a)
F
µ A (a)
(2)
+
X
N (a)≤x
x N (a)
F
µ A (a) .
The first term on the right-hand side is o(x) by Lemma 1, applied to f (a) = µ A (a); the assumptions of Lemma 1 are satisfied in view of Lemma 2 and the inequality
X
N (a)≤x
|µ A (a)| ≤ N A (x).
The second term on the right-hand side of (2) is o(x) by the identity X
N (a)≤x
x N (a)
F
µ A (a) = X
N (c)≤x
X
a|c
µ A (a) = N P \A (x) and Lemma 2. Hence
αx X
N (a)≤x
µ A (a)
N (a) = o(x) + o(x) = o(x)
and the proof of the Theorem has been finished.
EULER–VON MANGOLDT’S EQUATION
145
R e m a r k 1. In the case F = Q one can dispense with Axer’s theorem because of the straightforward estimation
X
n≤x
x n − x
n
µ A (n)
≤ X
n≤x
|µ A (n)| = o(x) and the proof becomes very easy!
R e m a r k 2. Using a standard sieve result ([2], p. 72, Corollary 2.3.1) and the above method of estimation one can obtain quantitative versions of the Theorem.
Proposition. Assume that A 1 ∪ A 2 = P , A 1 ∩ A 2 = ∅, and for each i = 1, 2 there exist δ i > 0 and M i ∈ R such that
X
p≤x, p∈A
i1
p ≥ δ i log log x − M i . Then
X
n≤x
µ A
i(n)
n = O((log x) − min(δ
1,δ
2) ).
As a very special corollary we obtain
Corollary. Let A i = {p ∈ P : p ≡ i mod 4} for i = 1, 3. Then X
n≤x
µ A
i(n)
n = O
1
√ log x
for i = 1, 3.
Acknowledgments. I want to thank an anonymous referee for his critical remarks and several improvements.
REFERENCES
[1] A. A x e r, Beitrag zur Kenntnis der zahlentheoretischen Funktionen µ(n) und λ(n), Prace Mat.-Fiz. 21 (1910), 65–95.
[2] H. H a l b e r s t a m and H. E. R i c h e r t, Sieve Methods, Academic Press, 1974.
[3] E. L a n d a u, ¨ Uber die zahlentheoretische Funktion µ(k), Wiener Sitzungsber. 112 (1903), 537–570.
[4] H. v o n M a n g o l d t, Beweis der Gleichung P
∞k=1
