User:Peter Luschny/RiemannLagarias

Riemann Hypothesis and
the Lagarias Formula.

KEYWORDS: Riemann Hypothesis, Lagarias Formula, Vandermast Numbers.

Concerned with sequences:  A057641, A094348, A181852.

This page is under construction!

Lagarias' criterion for the RH

The J. C. Lagarias criterion for the Riemann Hypothesis: A057641(n) ≥ 0.

A057641 := proc(n) local H, k;
H := add(1/k, k=1..n);
floor(H+exp(H)*log(H)) - numtheory[sigma](n) end:

Vandermast numbers

Matthew Vandermast's A094348:
Numbers n such that, for some numbers (j, k), j ≤ k, n is the smallest positive multiple of j of the first k positive integers.

Lagarias(n) = L(n) = A057641(n) =
0, 0, 1, 0, 4, 0, 7, 2, 7, 5, 13, 0, 17, 9, 12, 8, 23,
5, 27, 8, 21, 20, 34, 1, 33, 25, 30, 17, 46, 7, 50, 22,
40, 37, 46, 6, 62, 43, 50, 19, 70, 19, 74, 37, 46, 55,
82, 9, 79, 46, 70, 47, 95, 32, 83, 38, 81, 74, 107, 2,
112, 81, 76, 56, 102, 45, 125, 70

VandermastNumber(n) = V(n) = A094348(n) = 
1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 180, 
240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 
10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 
55440, 83160, 110880, 166320, 221760, 277200, 332640, 
360360, 498960, 554400, 665280, 720720

Lagarias(VandermastNumber(n)) = A181852(n) = 
A057641(A094348(n)) =
0, 0, 0, 0, 0, 1, 6, 9, 2, 14, 6, 23, 33, 34, 78, 114, 
105, 227, 281, 207, 492, 1536, 1667, 3036, 5155, 6502,
5206, 7682, 15861, 15396, 9051, 21295, 22160, 36300, 
58657, 71186, 81276, 91901, 165315, 154544, 199731, 
143828

Maple implementation

A094348_list := proc(N) 
local n,i,p,card,Max,A,S; 
A := 1;
for n from 2 to N do
   Max := 0; S := {$1..n};
   for i from 1 to N do
     card := nops(S intersect divisors(i));
     if Max < card then Max := card; A := A,i fi
   od
od;
{A} end:

A094348_list(120);
{1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120}

A working hypothesis

Conjecture: For n in the interval [V_i-1, V_i] min(L(V(i-1),L(V(i)) ≤ L(n).

This was numerically verified for 2 ≤ i ≤ 41.

Thus to verify the RH in the Lagarias form for n ≤ 720720 it is sufficient to check only 40 times the Lagarias criterion, i.e. to check 0 ≤  L(V(i)).

When I considered the first time the connection between the Vandermast numbers and the Lagarias criterion I formulated a slightly stronger form of the conjecture, namely that the polygonal chain which arises from the Vandermast numbers is a lower bound for the Lagarias values. This turned out not to be true. In the range below 720720 there is exactly one exception: 32760. I expect that there are more exceptions of this kind for greater n and I would like to find more of those integers as an exception of this kind is not accidental; it might point to new interesting relations.

A lower bound > 0 for L(V(n))?

If there is a simple lower bound > 0 for the values of the Lagarias formula at Vandermast numbers this would simplify things further since only values small in magnitude are of importance in the connection of the Lagarias formula with the Riemann Hypothesis. Note that the green line in the plot above simply is n/4.

See also the extended tables of A094348 and A181852.