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Revision History for A097878 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
A097878
Decimal expansion of Sum_{k>=1} k/prime(k)^4.
(history; published version)
#20 by Michael De Vlieger at Mon Nov 07 20:27:21 EST 2022
STATUS

proposed

approved

#19 by Jon E. Schoenfield at Mon Nov 07 19:50:52 EST 2022
STATUS

editing

proposed

#18 by Jon E. Schoenfield at Mon Nov 07 19:48:13 EST 2022
COMMENTS

Let M = 10^10, and let J be the number of primes < M, i.e., J = pi(M) = 455052511; then prime(455052512J+1) = 10000000019.

and it can be shown that the sum on the right-hand side is a value < 5*10^-22.

Summing the values of k/prime(k)^4 for all k <= J gives a lower boundto obtain

so an upper yields a lower bound on the infinite sum can be obtained as, and since the infinite sum is

Sum_{k>=1} k/prime(k)^4 = Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(k)^4,

it must be less than

= Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(M + 2*(k - J))^4,

< Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/(M + 2*(k - J))^4

which is less than

< 0.0944418581965049421842 + 5*10^-22 = 0.0944418581965049421847. (End),

which thus provides an upper bound on the infinite sum. (End)

Discussion
Mon Nov 07
19:49
Jon E. Schoenfield: I'll be happy to move the proof to a file under the Links, if that seems like a better idea.
19:50
Jon E. Schoenfield: It should not take too much computation to extend this by a few more digits.
#17 by Jon E. Schoenfield at Mon Nov 07 19:37:45 EST 2022
COMMENTS

The sum exceeds 0.09444185819650072 at prime(64287) = 804589, so not all digits listed in the Data are correct. - Jon E. Schoenfield, Feb 03 2018

From Jon E. Schoenfield, Nov 07 2022: (Start)

Let M = 10^10, and let J be the number of primes < M, i.e., J = pi(M) = 455052511; prime(455052512) = 10000000019.

Since prime(J+1) > M+2 and prime(k+1) - prime(k) >= 2 for all k > 1, it follows that, for all k > J,

prime(k) > M + 2*(k - J)

and thus

k/prime(k)^4 < k/(M + 2*(k - J))^4

so

Sum_{k>J} k/prime(k)^4 < Sum_{k>J} k/(M + 2*(k - J))^4

and the sum on the right-hand side is a value < 5*10^-22.

Summing the values of k/prime(k)^4 for all k <= J gives a lower bound

Sum_{k=1..J} k/prime(k)^4 = 0.0944418581965049421841...

so an upper bound on the infinite sum can be obtained as

Sum_{k>=1} k/prime(k)^4

= Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(k)^4

< Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/(M + 2*(k - J))^4

< 0.0944418581965049421842 + 5*10^-22 = 0.0944418581965049421847. (End)

#16 by Jon E. Schoenfield at Mon Nov 07 18:40:32 EST 2022
DATA

0, 9, 4, 4, 4, 1, 8, 5, 8, 1, 9, 6, 5, 0, 0, 7, 4, 9, 4, 2, 1, 8, 4

EXAMPLE

0.09444094441858196504942184...

EXTENSIONS

a(15)-a(17) corrected and a(18)-a(21) added by Jon E. Schoenfield, Nov 07 2022

STATUS

approved

editing

#15 by Michael De Vlieger at Tue Nov 01 13:49:05 EDT 2022
STATUS

proposed

approved

#14 by Jon E. Schoenfield at Sun Oct 30 14:39:26 EDT 2022
STATUS

editing

proposed

Discussion
Sun Oct 30
14:48
Amiram Eldar: So the last term in the data section, 1, is wrong?
14:56
Jon E. Schoenfield: Yes.
17:26
Jon E. Schoenfield: I’m sure the last 3 digits in the Data are wrong, but I don’t have a rigorous mathematical proof of their correct values.
21:01
Jon E. Schoenfield: At prime (66204), the sum exceeds 0.094441858196501, which shows that the last three terms in the Data are incorrect.
21:12
Jon E. Schoenfield: (Well, it doesn’t show that the last two terms are wrong, but it does show that the term before them is.) :-)
21:33
Jon E. Schoenfield: Okay, I thought of a way to rigorously prove some more of the digits I had given earlier, but it strikes me as a dumb way to do it: take the partial sum out to some fairly large index K, e.g., K = 10^7 (so prime(K) = 179424673), getting the partial sum 
S(K) = 0.09444185819650494214088867184 as a lower bound on the infinite sum; then note that prime(j) >= prime(K) + 2*(j-K) = f(j) for all j > K, so the tail of the summation T = Sum_{k>K} k/prime(k)^4 cannot exceed Sum_{k>K} k/f(k)^4, which can be evaluated exactly to get an upper bound for T, and add that to S(K) to get an upper bound for the infinite sum.
21:37
Jon E. Schoenfield: But I’m sure you could come up with something much better. I don’t know whether something involving the prime zeta function would yield an acceptably rigorous result or not here. I’m in over my head, giving up now.
Mon Oct 31
06:57
Amiram Eldar: I suggest that will remove the last 3 terms, and add an extension like 3 last term removed by ... You can also add your new lower bound as a comment.
Tue Nov 01
08:49
Jon E. Schoenfield: Ok, thanks, will try to get to that tonight or tomorrow.
#13 by Jon E. Schoenfield at Sun Oct 30 14:38:26 EDT 2022
COMMENTS

The sum exceeds 0.09444185819650072 at prime(64287) = 804589, so not all digits listed in the Data are correct. It appears to me that the actual limit is 0.094441858196504942184... - Jon E. Schoenfield, Feb 03 2018

STATUS

approved

editing

Discussion
Sun Oct 30
14:39
Jon E. Schoenfield: Unproven statement deleted.
#12 by Bruno Berselli at Fri Feb 09 03:38:28 EST 2018
STATUS

reviewed

approved

#11 by Joerg Arndt at Fri Feb 09 03:13:36 EST 2018
STATUS

proposed

reviewed

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