Abstract
AbstractWe present a new elementary algorithm that takes $$ \textrm{time} \ \ O_\epsilon \left( x^{\frac{3}{5}} (\log x)^{\frac{8}{5}+\epsilon } \right) \ \ \textrm{and} \ \textrm{space} \ \ O\left( x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right) $$
time
O
ϵ
x
3
5
(
log
x
)
8
5
+
ϵ
and
space
O
x
3
10
(
log
x
)
13
10
(measured bitwise) for computing $$M(x) = \sum _{n \le x} \mu (n),$$
M
(
x
)
=
∑
n
≤
x
μ
(
n
)
,
where $$\mu (n)$$
μ
(
n
)
is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $$O(x^{1/5} (\log x)^{5/3})$$
O
(
x
1
/
5
(
log
x
)
5
/
3
)
by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of $$x^{3/5} (\log x)^2 \log \log x$$
x
3
/
5
(
log
x
)
2
log
log
x
.
Funder
Alexander von Humboldt-Stiftung
European Research Council
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference21 articles.
1. Deléglise, M., Rivat, J.: Computing the summation of the Möbius function. Exp. Math. 5(4), 291–295 (1996)
2. Dress, F.: Fonction sommatoire de la fonction de Möbius; 1. Majorations expérimentales. Exp. Math. 2, 93–102 (1993)
3. Galway, W.F.: Dissecting a Sieve to Cut Its Need for Space. Algorithmic Number Theory (Leiden, 2000), Lecture Notes in Comput. Sci., pp. 297–312 (2000)
4. Helfgott, H.A.: The ternary Goldbach problem. Second preliminary version. Ann. Math. Stud. (to appear) https://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html
5. Helfgott, H.: An improved sieve of Eratosthenes. Math. Comput. 89, 333–350 (2020)