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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 908045, 11 pages
http://dx.doi.org/10.1155/2008/908045
Research Article

Generalized Newman Phenomena and Digit Conjectures on Primes

Departments of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

Received 2 June 2008; Accepted 26 August 2008

Academic Editor: Wee Teck Gan

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.

Abstract

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.

1. Introduction

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 [1], it follows that The case was studied in detail in [24].

So, from Coquet's theorem [3, 5] it follows that with a microscopic improvement [4]: and, moreover,

These estimates give the most exact modern limits of the so-called Newman phenomena. Note that Drmota and Skałba [6], 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

Using Theorem 1.2 and (1.5), one can estimate in (1.6). For example, one can prove that .

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),

Formulas (2.1)–(2.3) give an explicit expression for as a linear combination of the products of the form

Remark 2.2. On can extract (2.3) from a very complicated general Gelfond formula [1], however, we prefer to give an independent proof.

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 [1], 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 .

Thus, in (3.15), and (3.12) follows.

In case (2), let . For , put such that By (3.9) and (3.19), we have where .

Thus, according to (3.10) and taking into account that , we find while by (3.19) Now, in view of (3.21) and (3.11), and according to (3.14), (3.15), we obtain that where is defined by (3.13).

Notice that from simple arguments and according to (1.9),

Therefore, Now, by (3.2)-(3.4), for we have

Note that, according to (1.7) and (3.1), Thus, by (1.8) Thus, the theorem follows from (2.1).

4. Proof of Theorem 1.2

Select in (2.1) the summands which correspond to .

We have

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

Proof. Since then the corollary immediately follows from Theorems 1.1, 1.2.

5. On Newman Sum over Primes

In [7], 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.

Conjecture 5.2.

A heuristic proof of Conjecture 5.2 was given in [8]. 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 [8] 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

In conclusion, note that for , we have such that Thus, using Theorems 1.1, 1.2 in the form and inclusion-exclusion for , we find

Now, in view of (1.5), we obtain the following absolute result as an approximation of Conjectures 5.1, 5.2.

Theorem 5.4. For every prime number and sufficiently large , we have and, moreover,

References

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