[seqfan] Re: Generally known (to those who know of such things)
Jonathan Post
jvospost3 at gmail.com
Fri Nov 20 20:16:47 CET 2009
"there are more primes that are 3 mod 4 than 1 mod 4." -- keep
reading. Use keywords "prime race"
Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest
Unsolved Problem in Mathematics. New York: Penguin, pp. 125-126, 2004.
Sloane, N. J. A. Sequences A038691, A096628, A096629, and A096630 in
"The On-Line Encyclopedia of Integer Sequences."
Weisstein, Eric W. "Chebyshev Bias." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/ChebyshevBias.html
On Fri, Nov 20, 2009 at 10:52 AM, Victor S. Miller
<victorsmiller at gmail.com> wrote:
> I don't quite understand the notation in your sum:
>
> Do you mean that u(t) = sum(k=1 to t) k*p1[k]
>
> Where p1[k] is the k-th prime =1 mod 4?
>
> In any case this may be related to "Chebyshev's bias" there are more
> primes that are 3 mod 4 than 1 mod 4.
>
> Victor
>
> Sent from my iPhone
>
> On Nov 20, 2009, at 11:39 AM, "Meeussen Wouter \(bkarnd\)" <wouter.meeussen at vandemoortele.com
> > wrote:
>
>> I don't understand, again, why the primes (p1) of form 4n+1 differ in
>> behaviour from those (p3) of form 4n+3 in the following:
>>
>> u=Sum=1..p1; k (p1-k) ) is strictly increasing in function of p1,
>> while
>> v=Sum(k=1..p3; k (p3-k) ) is not (in function of p3).
>>
>> v is descending from v(37) to v(38)
>> {4, 28, 66, 190, 322, 558, 946, 1316, 1888, 2278, 2982, 3476, 3652,
>> 5768, 5992, 8636, 9170, 10008, 12382, 13366, 15698,
>> 16826, 20628, 21492, 22788, 26314, 26786, 32026, 33132, 37872, 39566,
>> 40752, 47892, 54114, 55608, 61766, 71082, 70464
>>
>> I checked 'u' way up to 10,000 and found no 'drops'.
>>
>>
>> Wouter.
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