Research Article | Open Access
Vladimir Shevelev, "Generalized Newman Phenomena and Digit Conjectures on Primes", International Journal of Mathematics and Mathematical Sciences, vol. 2008, Article ID 908045, 12 pages, 2008. https://doi.org/10.1155/2008/908045
Generalized Newman Phenomena and Digit Conjectures on Primes
We prove that the ratio of the Newman sum over numbers multiple of a fixed integer, which is not a multiple of 3, and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to infinity. We also discuss a connection of our results with a digit conjecture on primes.
Denote for , where is the number of 1's in the binary expansion of . Sum (1.1) is a Newman digit sum. From the fundamental paper of Gelfond , it follows that The case was studied in detail in [2–4].
These estimates give the most exact modern limits of the so-called Newman phenomena. Note that Drmota and Skałba , using a close function , proved that if is a multiple of 3, then for sufficiently large ,
In this paper, we study a general case for (in the cases of and , we have .
To formulate our results, put for where Directly, one can see that and thus, Below, we prove the following results.
Theorem 1.1. If , then
Theorem 1.2 (Generalized Newman phenomena). If is a multiple of 3, then
2. Explicit Formula for
We have Note that the interior sum has the form
Lemma 2.1. If , then where as usual .
Proof. Let ,
then by (2.2), which corresponds to (2.3) for .
Assuming that (2.3) is valid for every with , let us consider where is odd, , and . Let Notice that for , we have Therefore, Thus, by (2.3) and (2.4),
3. Proof of Theorem 1.1
Note that in (2.3) By Lemma 2.1, we have Furthermore, and, therefore, According to (3.2), let us estimate the product where by (2.1), Repeating arguments of , put Considering the function we have Note that for , where is the only positive root of the equation .
Show that either or, simultaneously, , and Indeed, let for a fixed values of and , Then, Now, distinguish two cases: (1) .
In case (1), and since , then Because of the condition , we have .
Notice that from simple arguments and according to (1.9),
4. Proof of Theorem 1.2
Select in (2.1) the summands which correspond to .
Since the chosen summands do not depend on and, for , the latter sum is empty, then we find
Further, the last double sum is estimated by the same way as in Section 3 such that
Remark 4.1. Notice that from elementary arguments it follows that if is a multiple of , then The latter expression is the value of in this case (see (1.9)).
Example 4.2. Let us find some such that for
Supposing that is multiple of 3 and using (1.4), we obtain that Therefore, putting in Theorem 1.2, we have Now, calculating and by (1.2) and (1.8), we find a required :
Corollary 4.3. For which is not a multiple of 3, denote the set of the positive integers not exceeding which are multiples of and not multiples of 3. Then,
In particular, for sufficiently large , we have
5. On Newman Sum over Primes
In , we put the following binary digit conjectures on primes.
Conjecture 5.1. For all , where the summing is over all primes not exceeding .
More precisely, by the observations, beginning with . Moreover, the following conjecture holds.
A heuristic proof of Conjecture 5.2 was given in . For a prime , denote the set of positive integers not exceeding for which is the least prime divisor. Show that the correctness of Conjectures 5.1 (for ) follows from the following very plausible statement, especially in view of the above estimates.
Conjecture 5.3. For sufficiently large , we have
Indeed, in the “worst case” (really is not satisfied), in which for all we have a decreasing but positive sequence of sums: Hence, the “balance condition” for odd numbers  must be ensured permanently by the excess of the odious primes. This explains Conjecture 5.1.
It is very interesting that for some primes the inequality (5.4), indeed, is satisfied for all . Such primes we call “resonance primes." Our numerous observations show that all resonance primes not exceeding are
Theorem 5.4. For every prime number and sufficiently large , we have and, moreover,
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Copyright © 2008 Vladimir Shevelev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.