User:Enrique Pérez Herrero/Riemann

NOTES ON RIEMANN HYPOTHESIS AND SEQUENCES

Robin Inequality

$\displaystyle \sigma (n)<e^{\gamma }n\log {\log {n}}$, for $\displaystyle n>5040$

• A094644: Continued Fraction for $e^{\gamma }$

$\displaystyle \psi (n)<e^{\gamma }n\log {\log {n}}$, for $\displaystyle n>30$ (Michel Planat)

Lagarias Inequality

• A181852: A057641(A094348(n))
• A057641: Floor(H(n)+exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.
• A094348: Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

Abundant numbers

• A005101: Abundant numbers (sum of divisors of n exceeds 2n).
• A006038: Odd primitive abundant numbers.

Highly Abundant Numbers

• A002093: Highly abundant numbers: sigma(n) > sigma(m) for all m < n.
• A192929: Least Highly Abundant Number with n distinct prime factors.

Superabundant Numbers (SA)

• A004394: Superabundant [or super-abundant] numbers: n such that ${\frac {\sigma (n)}{n}}>{\frac {\sigma (m)}{m}}$ for all $m<n$, $\sigma (n)$ being the sum of the divisors of n.

• Theorem: If there is any counterexample to Robin’s inequality, then the least such counterexample is a superabundant number.

Colossally Abundant Numbers (CA)

• A004490: Colossally abundant numbers: n for which there is a positive exponent epsilon such that sigma(n)/n^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that n attains the maximum value of sigma(n)/n^{1 + epsilon}.
• A073751: Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490.

Highly Composite Numbers

• A002182: Highly composite numbers, definition (1): where d(n), the number of divisors of n (A000005), increases to a record.

Deeply Composite Numbers

• A095848: Deeply composite numbers: numbers n where sigma_k(n) increases to a record for all sufficiently low values of k.

Superior Highly Composite Numbers

• A002201: Superior highly composite numbers: positive integers n for which there is an $e>0$ such that $d(n)/n^{e}\leq d(k)/k^{e}$ for all $k>1$, where the function $d(n)$ counts the divisors of n (A000005).
• A000705: n-th superior highly composite number A002201(n) is product of first n terms of this sequence.

A003418: Least Common Multiples of 1 through n

• A003418: a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, ..., n}.

Properties:

• if n is a prime power p^k, then A003418(p^k)=p*A003418(p^k-1), otherwise: A003418(n)=A003418(n-1).

• $A003418(n)=\prod _{k=2}^{n}{{\bigg (}1+(rad(k)-1)\cdot {\bigg \lfloor }{\frac {1}{\omega (k)}}{\bigg \rfloor }{\bigg )}}$

• $A003418(n)=\prod _{p\leq n}^{}{p^{\lfloor {\frac {\log {n}}{\log {p}}}\rfloor }}$

• $A003418(n)=\prod _{k=1}^{n}{e^{\Lambda (k)}}$

A003418: Mathematica Code

• Recursion with memorization technique, can calculate 10000 terms in less then 0.2 seconds

A003418[0]:= 1;
A003418[1]:= 1;
A003418[n_]:=A003418[n]=LCM[n,A003418[n-1]];

A096179: Triangle read by rows: T(n,k) is the smallest positive integer having at least k of the first n positive integers as divisors.

Properties:

• T(p,k)=T(p-1,k)
• T(p,p)=p*T(p-1,p-1)

A096179: Mathematica Code

(* Triangular *) 
A096179[n_, k_]:=Min[LCM@@@Subsets[Range[n], {k}]]; 
A002024[n_]:=Floor[1/2+Sqrt[2*n]]; 
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]], 2]; 
(* Linear *) 
A096179[n_]:=A096179[n]=A096179[A002024[n], A002260[n]];

Links:

• [1] J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Amer. Math. Monthly, 109 (2002), 534-543
• [2] Keith Briggs, Abundant Numbers and the Riemann Hypothesis, Experimental Mathematics, Vol. 15 (2006), No. 2
• [3] Keith Briggs, Notes on the Riemann hypothesis and abundant numbers, 2005 April 21
• [4] Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers". Transactions of the American Mathematical Society 56 (3): 448–469. doi:10.2307/1990319. MR0011087. http://jstor.org/stable/1990319.
• [5] Erdös, Pál, On highly composite numbers, J. London Math. Soc. 19, 130-133 (1944).
• [6] YoungJu Choie; Nicolas Lichiardopol; Pieter Moree; Patrick Solé, "On Robin’s criterion for the Riemann hypothesis", Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 357-372
• [7] Patrick Solé, Michel Planat, "Robin inequality for 7-free integers", arXiv:1012.0671v1, Dec, 3, 2010.
• [8] Amir Akbary and Zachary Friggstad, Superabundant Numbers and the Riemann Hypothesis
• [9] Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239v2