[math-fun] Alan Adler, of Aerobie fame,
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
One of my favorites is Brun's theorem that shows that the sum of reciprocals of the twin primes converges to a value now known as Brun's constant: http://oeis.org/A065421 Victor PS. To my mind the greatest thing that Alan Adler did was to invent the Aeropress. If you don't know about it, it's an incredible way of making great coffee. On Sun, Jun 2, 2013 at 12:04 AM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
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As someone whose daily mainstay is instant coffee in microwaved milk, I'm amazed at the continuing evolution of coffee culture: I've recently had to look up the word 'barista'. At any rate, I'm pleased that the Aeropress doesn't involve any throwing and catching. On Jun 2, 2013, at 10:03 AM, Victor Miller <victorsmiller@gmail.com> wrote:
To my mind the greatest thing that Alan Adler did was to invent the Aeropress. If you don't know about it, it's an incredible way of making great coffee.
Just in case anyone searches "barista" and finds "David Makin" that's not me, but a world-leading barista from down-under ;) On 2 Jun 2013, at 15:18, Hans Havermann wrote:
As someone whose daily mainstay is instant coffee in microwaved milk, I'm amazed at the continuing evolution of coffee culture: I've recently had to look up the word 'barista'. At any rate, I'm pleased that the Aeropress doesn't involve any throwing and catching.
On Jun 2, 2013, at 10:03 AM, Victor Miller <victorsmiller@gmail.com> wrote:
To my mind the greatest thing that Alan Adler did was to invent the Aeropress. If you don't know about it, it's an incredible way of making great coffee.
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
http://mathworld.wolfram.com/PrimeProducts.html Lots of fun stuff here, like prod(i=1, inf, p_i^s) = (2pi)^2s On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Also, the following paper "Zeta Expansions of Classical Constants" (for some reason never published): http://algo.inria.fr/flajolet/Publications/landau.ps On Sun, Jun 2, 2013 at 10:58 AM, Mike Stay <metaweta@gmail.com> wrote:
http://mathworld.wolfram.com/PrimeProducts.html
Lots of fun stuff here, like prod(i=1, inf, p_i^s) = (2pi)^2s
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Wow, I just found a smearoxed 241(!) page paper, On products and series involving prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine. On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
Unfortunately, Glaisher's very next example is a product from "p=2" to "p=∞" of an f(s,p), "where p is any prime, and s = +1 or -1 according as p ≡ 1 or ≡ 2 mod 3". What's s(3)?? Numerically, it seems Product[Prime[n]^(1/(-1 + (3 - 2*Mod[Prime[n], 3])*Prime[n])), {n, 1, ∞}] == (3*3^(5/8)*Gamma[1/3]^6)/(E^EulerGamma*(16*Pi^4)), although NLimit lingers in bovoparturition. Note that for these products to converge, the number of 3k+1 primes <n must stay roughly even with the # of 3k+2 as n->∞. Likewise 4k+1 and 4k+3. --rwg On Sun, Jun 2, 2013 at 1:55 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow, I just found a smearoxed 241(!) page paper, On products and series involving prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine.
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
Ah, later, during the derivation, Glaisher clarifies: 1 2 1/6 1/12 1/18 1/30 1/36 6 EllipticK[- (2 - Sqrt[3])] 7 13 19 31 37 4 -------------------------------- ... = ----------------------------- 1/6 1/12 1/18 1/24 1/30 EulerGamma 2 5 11 17 23 29 E Pi where EllipticK[1/4 (2 - Sqrt[3])] == (3^(1/4) Gamma[1/3]^3)/(4 2^(1/3) π) (Apparently, K was considered a special value of Gamma whenever possible.) Note how Mma automatically collects all the twin primes: Rule @@ %152 /. oo -> 14 43 1/42 31 1/30 19 1/18 13 1/12 7 1/6 1/36 (--) (--) (--) (--) (-) 37 41 29 17 11 5 ------------------------------------------------- -> 1/24 23 1 2 6 EllipticK[- (2 - Sqrt[3])] 4 ----------------------------- EulerGamma 2 E Pi Convergence is less smooth than this would suggest--even nonmonotonic, I think. Which explains why yesterday's NLimit of the infinite product is still running. The bunchiness of the primes is nonlethal: In[126]:= NLimit[Product[1 - Prime[k]^-2, {k, n}], n -> ∞] Out[126]= 0.607927 In[127]:= % - 6/π^2 Out[127]= -2.39218*10^-9 took a couple of minutes. --rwg On Sun, Jun 2, 2013 at 8:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
Unfortunately, Glaisher's very next example is a product from "p=2" to "p=∞" of an f(s,p), "where p is any prime, and s = +1 or -1 according as p ≡ 1 or ≡ 2 mod 3". What's s(3)?? Numerically, it seems
Product[Prime[n]^(1/(-1 + (3 - 2*Mod[Prime[n], 3])*Prime[n])), {n, 1, ∞}] == (3*3^(5/8)*Gamma[1/3]^6)/(E^EulerGamma*(16*Pi^4)),
although NLimit lingers in bovoparturition.
Note that for these products to converge, the number of 3k+1 primes <n must stay roughly even with the # of 3k+2 as n->∞. Likewise 4k+1 and 4k+3. --rwg
On Sun, Jun 2, 2013 at 1:55 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow, I just found a smearoxed 241(!) page paper, On products and series involving prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine.
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
Here's a nice one (a page or two later): Product[Prime[n]^(1/(Prime[n]^2 - 1)), {n, ∞]}]==E^-(Zeta'[2]/Zeta[2]) . Unfortunately, Mma turns this into a big mess involving the useless symbol Glaisher, which is about as handy as if it arbitrarily gronked zeta[3] to be Apery. --rwg On Mon, Jun 3, 2013 at 12:41 PM, Bill Gosper <billgosper@gmail.com> wrote:
Ah, later, during the derivation, Glaisher clarifies: 1 2 1/6 1/12 1/18 1/30 1/36 6 EllipticK[- (2 - Sqrt[3])] 7 13 19 31 37 4 -------------------------------- ... = ----------------------------- 1/6 1/12 1/18 1/24 1/30 EulerGamma 2 5 11 17 23 29 E Pi
where EllipticK[1/4 (2 - Sqrt[3])] == (3^(1/4) Gamma[1/3]^3)/(4 2^(1/3) π)
(Apparently, K was considered a special value of Gamma whenever possible.)
Note how Mma automatically collects all the twin primes: Rule @@ %152 /. oo -> 14
43 1/42 31 1/30 19 1/18 13 1/12 7 1/6 1/36 (--) (--) (--) (--) (-) 37 41 29 17 11 5 ------------------------------------------------- -> 1/24 23
1 2 6 EllipticK[- (2 - Sqrt[3])] 4 ----------------------------- EulerGamma 2 E Pi
Convergence is less smooth than this would suggest--even nonmonotonic, I think. Which explains why yesterday's NLimit of the infinite product is still running. The bunchiness of the primes is nonlethal:
In[126]:= NLimit[Product[1 - Prime[k]^-2, {k, n}], n -> ∞]
Out[126]= 0.607927
In[127]:= % - 6/π^2
Out[127]= -2.39218*10^-9
took a couple of minutes. --rwg
On Sun, Jun 2, 2013 at 8:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
Unfortunately, Glaisher's very next example is a product from "p=2" to "p=∞" of an f(s,p), "where p is any prime, and s = +1 or -1 according as p ≡ 1 or ≡ 2 mod 3". What's s(3)?? Numerically, it seems
Product[Prime[n]^(1/(-1 + (3 - 2*Mod[Prime[n], 3])*Prime[n])), {n, 1, ∞}] == (3*3^(5/8)*Gamma[1/3]^6)/(E^EulerGamma*(16*Pi^4)),
although NLimit lingers in bovoparturition.
Note that for these products to converge, the number of 3k+1 primes <n must stay roughly even with the # of 3k+2 as n->∞. Likewise 4k+1 and 4k+3. --rwg
On Sun, Jun 2, 2013 at 1:55 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow, I just found a smearoxed 241(!) page paper, On products and series involving prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine.
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com>wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
On Wed, Jun 5, 2013 at 2:25 AM, Bill Gosper <billgosper@gmail.com> wrote:
Here's a nice one (a page or two later): Product[Prime[n]^(1/(Prime[n]^2 - 1)), {n, ∞]}]==E^-(Zeta'[2]/Zeta[2]) . Unfortunately, Mma turns this into a big mess involving the useless symbol Glaisher, which is about as handy as if it arbitrarily gronked zeta[3] to be Apery. --rwg
E.g, In[10]:= FunctionExpand[Zeta'[2]] Out[10]= 1/6 \[Pi]^2 (EulerGamma + Log[2] - 12 Log[Glaisher] + Log[\[Pi]]) [...]
Numerically testing, In[7]:= N[Product[Prime[n]^(Prime[n]^2 - 1)^-1, {n, \[Infinity]}]] - E^-(Zeta'[2]/Zeta[2]) During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[15.]. >> During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[14.]. >> During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[13.]. >> During evaluation of In[7]:= General::stop: Further output of Prime::intpp will be suppressed during this calculation. >> Out[7]= -0.0084166 Children should be shielded from software that claims 15. is not a positive integer. A stronger test: NLimit[Product[Prime[n]^(Prime[n]^2 - 1)^-1, {n, oo}], oo -> \[Infinity]] - E^-(Zeta'[2]/Zeta[2]) returned ~10^-7 after a couple of days. I had selected the output, but not copied it into the clipboard, when I switched notebooks for a trivial search. The third time I clicked Next, the whole front end crashed. --rwg
(Repeated from below:) Caution: Product[Prime[n]^(1/(Prime[n]* (3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine. WRI just informed me they fixed this in Mma 10. I don't trust this transcription, but Mod[Prime[n], 3] 1/(1 + (-1) Prime[n]) Product[Prime[n] , EulerGamma 4 16 E Pi {n, oo}] -> ------------------ 1/4 1 6 3 3 Gamma[-] 3 seems to work mathematically (with crash-provoking ∞ for oo). Is this known to converge? I doubt Glaisher's derivation provides proof. This suggests a bunch of convergence puzzles of the form Sum (-1)^Axxx(n)/f(n), where Axxx is some integer sequence and f is some slowly growing function like n or log(1+n) or log(1+log(1+n)) ... . Presumably a sufficiently gradual f could betray bunchiness mod 2 in something like Mod[Prime[n],3] = A039701 <http://oeis.org/A039701>. Glaisher used a master theorem: In[280]:= Product[n^(Sin[2*n*\[Pi]*\[Mu]]/n),{n,\[Infinity]}]->(Sqrt[Sin[\[Pi]*\[Mu]]]*\[CapitalGamma][\[Mu]]/(2^(1/2-\[Mu])*\[Pi]^(1-\[Mu])*E^((1/2-\[Mu])*EulerGamma)))^\[Pi] Notice Mma's evaluation of the lhs: Out[280]= E^(-(1/2) I ((PolyLog^(1,0))[1,E^(-2 I \[Pi] \[Mu])]-(PolyLog^(1,0))[1,E^(2 I \[Pi] \[Mu])]))->(2^(-(1/2)+\[Mu]) E^(-EulerGamma (1/2-\[Mu])) \[Pi]^(-1+\[Mu]) Sqrt[Sin[\[Pi] \[Mu]]] \[CapitalGamma][\[Mu]])^\[Pi] That's the derivative wrt the *order* of the polylog, of whose numerics it seems capable. --rwg On Mon, Jun 3, 2013 at 12:41 PM, Bill Gosper <billgosper@gmail.com> wrote:
Ah, later, during the derivation, Glaisher clarifies: 1 2 1/6 1/12 1/18 1/30 1/36 6 EllipticK[- (2 - Sqrt[3])] 7 13 19 31 37 4 -------------------------------- ... = ----------------------------- 1/6 1/12 1/18 1/24 1/30 EulerGamma 2 5 11 17 23 29 E Pi
where EllipticK[1/4 (2 - Sqrt[3])] == (3^(1/4) Gamma[1/3]^3)/(4 2^(1/3) π)
(Apparently, K was considered a special value of Gamma whenever possible.)
Note how Mma automatically collects all the twin primes: Rule @@ %152 /. oo -> 14
43 1/42 31 1/30 19 1/18 13 1/12 7 1/6 1/36 (--) (--) (--) (--) (-) 37 41 29 17 11 5 ------------------------------------------------- -> 1/24 23
1 2 6 EllipticK[- (2 - Sqrt[3])] 4 ----------------------------- EulerGamma 2 E Pi
Convergence is less smooth than this would suggest--even nonmonotonic, I think. Which explains why yesterday's NLimit of the infinite product is still running. The bunchiness of the primes is nonlethal:
In[126]:= NLimit[Product[1 - Prime[k]^-2, {k, n}], n -> ∞]
Out[126]= 0.607927
In[127]:= % - 6/π^2
Out[127]= -2.39218*10^-9
took a couple of minutes. --rwg
On Sun, Jun 2, 2013 at 8:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
Unfortunately, Glaisher's very next example is a product from "p=2" to "p=∞" of an f(s,p), "where p is any prime, and s = +1 or -1 according as p ≡ 1 or ≡ 2 mod 3". What's s(3)?? Numerically, it seems
Product[Prime[n]^(1/(-1 + (3 - 2*Mod[Prime[n], 3])*Prime[n])), {n, 1, ∞}] == (3*3^(5/8)*Gamma[1/3]^6)/(E^EulerGamma*(16*Pi^4)),
although NLimit lingers in bovoparturition.
Note that for these products to converge, the number of 3k+1 primes <n must stay roughly even with the # of 3k+2 as n->∞. Likewise 4k+1 and 4k+3. --rwg
On Sun, Jun 2, 2013 at 1:55 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow, I just found a smearoxed 241(!) page paper,
http://digreg.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=reg&ci=QJPAM&id=ART&s...
On products and series involving
prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine.
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg
participants (5)
-
Bill Gosper -
Dave Makin -
Hans Havermann -
Mike Stay -
Victor Miller