[seqfan] Re: Primes arising in Newman digit sums

[Vladimir Shevelev]

Dear Charles,

Your question is open and very difficult. In the same paper, M. Drmota and M. Skalba proved for Newman sum S_{m,0}(x)=Sum{0<=n<x, n==0 (mod m)} (-1)^s(n),  where s(n)=A000120(n),
that if m is multiple of 3, then S_{m,0}(x)>0 for x>=x_0.
Note that an estimate of x_0(m) is also a difficult problem. Using method of trigonometrical sums, I
in paper http://www.hinduwi.com/jornals/ijmms/2008/908045html , in particular, get an algorithm
for such estimates. As an example, I found that if m=21, then x_0(21)<e^{909}.

Best regards,
Vladimir
----- Original Message -----
From: Charles Greathouse <charles.greathouse at case.edu>
Date: Wednesday, November 17, 2010 18:43
Subject: [seqfan]  Primes arising in Newman digit sums
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Consider sequence A131342: Primes arising in Newman digit 
> sums.  The comment
> 
> %C A remarkable result of M. Drmota and M. Skalba: the only 
> primes p
> =< 1000 satisfying S_p,0(n) > 0 (at least for sufficiently 
> large n)
> are 3, 5, 17, 43, 257, 683.
> 
> taken from
> http://arXiv.org/pdf/0709.3821
> 
> seems to suggest that the sequence should be finite and full 
> (and also
> base).  But instead it has keyword:more.  Do I 
> misunderstand a special
> case for the general one (in which case the sequence, I hope, 
> can be
> clarified), or are the keywords wrong?
> 
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
> 
> 
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> 

 Shevelev Vladimir‎