Xjyuk

Why do mean value theorems have open interval for differentiablity while closed for continuity?

Why doesn't Anakin's lightsaber explode when it's chopped in half on Geonosis?

Remove intersect line for one circle using venndiagram2sets

What is the standard dungeon scale at the table?

3D-Plot with an inequality condition for parameter values

Are there any double stars that I can actually see orbit each other?

Too many spies!

What's the phrasal verb for carbonated drinks exploding out of the can after being shaken?

Are L-functions uniquely determined by their values at negative integers?

Krazy language in Krazy Kat, 25 July 1936

Adding a vertical line at the right end of the horizontal line in frac

Behavior of the zero and negative/sign flags on classic instruction sets

Won 50K! Now what should I do with it

Can a pizza stone be fixed after soap has been used to clean it?

Is it rude to tell recruiters I would only change jobs for a better salary?

Commutator subgroup of Heisenberg group.

Can someone explain this logical statement?

Why hasn't the U.S. government paid war reparations to any country it attacked?

Did the Shuttle's rudder or elevons operate when flown on its carrier 747?

Connect neutrals together in 3-gang box (load side) with 3x 3-way switches?

How did John Lennon tune his guitar

Why linear regression uses "vertical" distance to the best-fit-line, instead of actual distance?

Why is "dark" an adverb in this sentence?

Are villager price increases due to killing them temporary?

Relationship between GCD, LCM and the Riemann Zeta function

Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?Calculating the Zeroes of the Riemann-Zeta functionWhat is so interesting about the zeroes of the Riemann $zeta$ function?How to evaluate a zero of the Riemann zeta function?Is there any relationship between the Riemann z function and strange attractors?Relationship between Riemann Zeta function and Prime zeta functionQuestion on Integral Transform Related to Riemann Zeta Function $zeta(s)$Relation of prime indexed series with Riemann zeta functionRiemann zeta for real argument between 0 and 1 using Mellin, with short asymptotic expansionReference request on the Riemann Zeta function

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

5

3

$begingroup$

Let $zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)textlcm(k,i)bigg)^s approx zeta(s+1)
$$

or equivalently

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)^2kibigg)^s approx zeta(s+1)
$$

A few values of $s$, LHS and the RHS are given below

$$(3,1.221,1.202)$$
$$(4,1.084,1.0823)$$
$$(5,1.0372,1.0369)$$
$$(6,1.01737,1.01734)$$
$$(7,1.00835,1.00834)$$
$$(9,1.00494,1.00494)$$
$$(19,1.0000009539,1.0000009539)$$

Question: Is the LHS asymptotic to $zeta(s+1)$ ?

number-theory elementary-number-theory summation prime-numbers analytic-number-theory

share|cite|improve this question

edited 8 hours ago

Nilotpal Kanti Sinha

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

$endgroup$

add a comment | 

5

3

$begingroup$

Let $zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)textlcm(k,i)bigg)^s approx zeta(s+1)
$$

or equivalently

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)^2kibigg)^s approx zeta(s+1)
$$

A few values of $s$, LHS and the RHS are given below

$$(3,1.221,1.202)$$
$$(4,1.084,1.0823)$$
$$(5,1.0372,1.0369)$$
$$(6,1.01737,1.01734)$$
$$(7,1.00835,1.00834)$$
$$(9,1.00494,1.00494)$$
$$(19,1.0000009539,1.0000009539)$$

Question: Is the LHS asymptotic to $zeta(s+1)$ ?

number-theory elementary-number-theory summation prime-numbers analytic-number-theory

share|cite|improve this question

edited 8 hours ago

Nilotpal Kanti Sinha

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

$endgroup$

add a comment | 

$begingroup$

Let $zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)textlcm(k,i)bigg)^s approx zeta(s+1)
$$

or equivalently

$$
frac1nsum_k = 1^nsum_i = 1^k bigg(fracgcd(k,i)^2kibigg)^s approx zeta(s+1)
$$

A few values of $s$, LHS and the RHS are given below

$$(3,1.221,1.202)$$
$$(4,1.084,1.0823)$$
$$(5,1.0372,1.0369)$$
$$(6,1.01737,1.01734)$$
$$(7,1.00835,1.00834)$$
$$(9,1.00494,1.00494)$$
$$(19,1.0000009539,1.0000009539)$$

Question: Is the LHS asymptotic to $zeta(s+1)$ ?

number-theory elementary-number-theory summation prime-numbers analytic-number-theory

share|cite|improve this question

edited 8 hours ago

Nilotpal Kanti Sinha

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

$endgroup$

share|cite|improve this question

edited 8 hours ago

Nilotpal Kanti Sinha

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

share|cite|improve this question

edited 8 hours ago

Nilotpal Kanti Sinha

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

asked 9 hours ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

5,37922 gold badges1616 silver badges4242 bronze badges

1 Answer
1

active

oldest

votes

6

$begingroup$

With $(A,B) = (ga,gb), gcd(a,b)=1$ then

$$sum_A,B, gcd(A,B) le G fracgcd(A,B)^slcm(A,B)^s = sum_gle G sum_a,b, gcd(a,b)=1fracgcd(ag,bg)^slcm(ag,bg)^s$$

$$= sum_gle G sum_a,b, gcd(a,b)=1fracg^s(abg)^s = G sum_a,b, gcd(a,b)=1frac1(ab)^s$$
$$ = G sum_d mu(d)sum_u,vfrac1(d^2uv)^s= G (sum_d mu(d)d^-2s)(sum_uu^-s)(sum_v v^-s) = G fraczeta(s)^2zeta(2s)$$

As $s to infty$ it $to G zeta(s)^2 approx Gzeta(s+1)$

share|cite|improve this answer

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3293112%2frelationship-between-gcd-lcm-and-the-riemann-zeta-function%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

6

$begingroup$

With $(A,B) = (ga,gb), gcd(a,b)=1$ then

$$sum_A,B, gcd(A,B) le G fracgcd(A,B)^slcm(A,B)^s = sum_gle G sum_a,b, gcd(a,b)=1fracgcd(ag,bg)^slcm(ag,bg)^s$$

$$= sum_gle G sum_a,b, gcd(a,b)=1fracg^s(abg)^s = G sum_a,b, gcd(a,b)=1frac1(ab)^s$$
$$ = G sum_d mu(d)sum_u,vfrac1(d^2uv)^s= G (sum_d mu(d)d^-2s)(sum_uu^-s)(sum_v v^-s) = G fraczeta(s)^2zeta(2s)$$

As $s to infty$ it $to G zeta(s)^2 approx Gzeta(s+1)$

share|cite|improve this answer

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

$endgroup$

add a comment | 

6

$begingroup$

With $(A,B) = (ga,gb), gcd(a,b)=1$ then

$$sum_A,B, gcd(A,B) le G fracgcd(A,B)^slcm(A,B)^s = sum_gle G sum_a,b, gcd(a,b)=1fracgcd(ag,bg)^slcm(ag,bg)^s$$

$$= sum_gle G sum_a,b, gcd(a,b)=1fracg^s(abg)^s = G sum_a,b, gcd(a,b)=1frac1(ab)^s$$
$$ = G sum_d mu(d)sum_u,vfrac1(d^2uv)^s= G (sum_d mu(d)d^-2s)(sum_uu^-s)(sum_v v^-s) = G fraczeta(s)^2zeta(2s)$$

As $s to infty$ it $to G zeta(s)^2 approx Gzeta(s+1)$

share|cite|improve this answer

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

$endgroup$

add a comment | 

$begingroup$

With $(A,B) = (ga,gb), gcd(a,b)=1$ then

$$sum_A,B, gcd(A,B) le G fracgcd(A,B)^slcm(A,B)^s = sum_gle G sum_a,b, gcd(a,b)=1fracgcd(ag,bg)^slcm(ag,bg)^s$$

$$= sum_gle G sum_a,b, gcd(a,b)=1fracg^s(abg)^s = G sum_a,b, gcd(a,b)=1frac1(ab)^s$$
$$ = G sum_d mu(d)sum_u,vfrac1(d^2uv)^s= G (sum_d mu(d)d^-2s)(sum_uu^-s)(sum_v v^-s) = G fraczeta(s)^2zeta(2s)$$

As $s to infty$ it $to G zeta(s)^2 approx Gzeta(s+1)$

share|cite|improve this answer

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

$endgroup$

share|cite|improve this answer

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges

answered 8 hours ago

reunsreuns

23.6k22 gold badges1414 silver badges6161 bronze badges