OFFSET
3,1
COMMENTS
Reciprocals of the constants determining the tail of limiting distribution of quadratic Weyl sums with rational parameters.
LINKS
Francesco Cellarosi and Tariq Osman, Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters, arXiv:2210.09838 [math.NT], 2022-2023.
FORMULA
Let DedekindPsi(m) = m * Product{p|m, p prime} (1 + 1/p).
Write n = m*2^k with odd m and k >= 0.
If k = 0 or k = 1, then const(n) = 2/DedekindPsi(m).
If k > 1, then const(n) = 1/(2^(k-1)*DedekindPsi(m)).
const(1) = const(2) = 1/2 and for all n >= 3, 1/const(n) is an integer.
a(n) = 1/const(n) for n >= 3.
From Peter Luschny, Oct 26 2022: (Start)
a(n) = (DedekindPsi(OddPart(n))/2) * [4 divides n ? EvenPart(n) : 1] = (A001615(A000265(n))/2) * [4 divides n ? A006519(n) : 1].
Define: k in S <=> k is a number that divides DedekindPsi(k) and its even part is greater than 2. Then sequence(k/12 for k in S) = sequence(a(k)/8 for k in S) = A003586. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 21/(8*Pi^2) = 0.265968... . - Amiram Eldar, Dec 09 2023
EXAMPLE
a(3) = DedekindPsi(3)/2 = 4/2 = 2.
a(4) = 2^(2-1)*DedekindPsi(1) = 2*1 = 2.
a(5) = DedekindPsi(5)/2 = 6/2 = 3.
a(6) = DedekindPsi(3)/2 = 2.
a(16) = 2^(4-1)*DedekindPsi(1) = 8*1 = 8.
a(120) = 2^(3-1)*DedekindPsi(15) = 4*24 = 96.
MAPLE
alias(DedekindPsi = A001615):
A358015 := proc(n) local k, h; k := padic[ordp](n, 2); h := 2^k;
DedekindPsi(n/h)/2; if h > 2 then %*h else % fi end:
seq(A358015(n), n = 3..75); # Peter Luschny, Oct 25 2022
MATHEMATICA
DedekindPsi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}];
const[n_]:=If[OddQ[n] || (Mod[n, 4] == 2), 2/DedekindPsi[n/2^IntegerExponent[n, 2]], 1/(2^(IntegerExponent[n, 2]-1) DedekindPsi[n/2^IntegerExponent[n, 2]])]
a[n_]:=1/const[n]
PROG
(SageMath)
from sage.arith.misc import dedekind_psi
def A358015(n):
k = valuation(n, 2)
j = k if mod(n, 4) == 0 else 0
return dedekind_psi(n*2^(-k))*2^(j-1)
print([A358015(n) for n in range(3, 76)]) # Peter Luschny, Oct 26 2022
(Python)
from math import prod
from sympy import primefactors
def A358015(n):
s =(m:=n>>(k:=(~n & n-1).bit_length()))//prod(q:=primefactors(m))*prod(p+1 for p in q)
return s>>1 if n&3 else s<<k-1 # Chai Wah Wu, Oct 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
F Cellarosi, Oct 24 2022
STATUS
approved