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The Robin Inequality for 7-Free Integers
-
Patrick Solé
and
Michel Planat
Published/Copyright:
March 27, 2012
Abstract.
Recall that an integer is 
-free if and only if it is not divisible by 
for some prime 
. We give a method to check Robin's inequality, 
, for 
-free integers 
and apply it for 
.
We introduce 
, a generalization of the Dedekind 
function defined for any integer 
, by
If 
is 
-free, then the sum of divisor function 
is 
. We characterize the champions for 
,
as primorial numbers. Define the ratio 
. We prove that, for all 
, there exists an integer 
such that we have 
for 
, where 
. Further, by combinatorial arguments, this can be extended to 
for all
such that 
. This yields Robin's inequality for 
. For 
varying slowly with 
, we also derive 
.
Received: 2010-12-01
Revised: 2011-08-24
Accepted: 2011-10-11
Published Online: 2012-03-27
Published in Print: 2012-April
© 2012 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
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- Odd Catalan Numbers Modulo
- Variations of the Poincaré Map
- Diophantine Equations of Matching Games I
- Norm Euclidean Quaternionic Orders
- A New Proof of Winquist's Identity
- Counting Depth Zero Patterns in Ballot Paths
- Codes Associated with and Power Moments of Kloosterman Sums
- Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime
- Generalized Nonaveraging Integer Sequences
- The Robin Inequality for 7-Free Integers
- On 3-adic Valuations of Generalized Harmonic Numbers
Keywords for this article
Dedekind Function;
Robin's Inequality;
Riemann
Hypothesis;
Primorial Numbers
Articles in the same Issue
- Masthead
- Odd Catalan Numbers Modulo
- Variations of the Poincaré Map
- Diophantine Equations of Matching Games I
- Norm Euclidean Quaternionic Orders
- A New Proof of Winquist's Identity
- Counting Depth Zero Patterns in Ballot Paths
- Codes Associated with and Power Moments of Kloosterman Sums
- Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime
- Generalized Nonaveraging Integer Sequences
- The Robin Inequality for 7-Free Integers
- On 3-adic Valuations of Generalized Harmonic Numbers
