Relationship between GCD, LCM and the Riemann Zeta functionValues 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
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;
$begingroup$
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
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 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
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 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
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
$endgroup$
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
number-theory elementary-number-theory summation prime-numbers analytic-number-theory
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
edited 8 hours ago
Nilotpal Kanti Sinha
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
asked 9 hours ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
5,3792 gold badges16 silver badges42 bronze badges
5,3792 gold badges16 silver badges42 bronze badges
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 $to G zeta(s)^2 approx Gzeta(s+1)$
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 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
);
);
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
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
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $to G zeta(s)^2 approx Gzeta(s+1)$
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 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 $to G zeta(s)^2 approx Gzeta(s+1)$
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 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 $to G zeta(s)^2 approx Gzeta(s+1)$
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 bronze badges
$endgroup$
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 $to G zeta(s)^2 approx Gzeta(s+1)$
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 bronze badges
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 bronze badges
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 bronze badges
answered 8 hours ago
reunsreuns
23.6k2 gold badges14 silver badges61 bronze badges
23.6k2 gold badges14 silver badges61 bronze badges
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
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
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
