Generalized Newman Phenomena and Digit Conjectures on Primes

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 𝑥,𝑚∈ℕ, 𝑆_𝑚(𝑥)=_{0≤𝑛<𝑥,𝑛≡0(mod𝑚)}(−1)^𝑠(𝑛), (1.1) where 𝑠(𝑛) is the number of 1's in the binary expansion of 𝑛. Sum (1.1) is a Newman digit sum. From the fundamental paper of Gelfond [1], it follows that 𝑆_𝑚(𝑥)=𝑂(𝑥^𝜆),𝜆=ln3ln4. (1.2) The case 𝑚=3 was studied in detail in [2–4].

So, from Coquet's theorem [3, 5] it follows that −13+2√3𝑥^𝜆≤𝑆₃1(3𝑥)≤3+553(3)65^𝜆𝑥^𝜆 (1.3) with a microscopic improvement [4]: 2√3𝑥^𝜆≤𝑆₃(3𝑥)≤553(3)65^𝜆𝑥^𝜆,𝑥≥2, (1.4) and, moreover, 𝑥⌊2(6)^𝜆⌋≤𝑆₃(𝑥)≤⌈553(𝑥)65^𝜆⌉. (1.5)

These estimates give the most exact modern limits of the so-called Newman phenomena. Note that Drmota and Skałba [6], using a close function (𝑆_𝑚^(𝑚)(𝑥)), proved that if 𝑚 is a multiple of 3, then for sufficiently large 𝑥, 𝑆_𝑚(𝑥)>0,𝑥≥𝑥₀(𝑚). (1.6)

In this paper, we study a general case for 𝑚≥5 (in the cases of 𝑚=2 and 𝑚=4, we have |𝑆_𝑚(𝑛)|≤1).

To formulate our results, put for 𝑚≥5,𝜆_𝑚=1+log₂𝑏_𝑚,𝜇_𝑚=22𝑏_𝑚,−1 (1.7)(1.8) where 𝑏²_𝑚=𝜋sin(33(1+𝑚))(√3−sin(𝜋33(1+𝑚𝜋))),if𝑚≡0(mod3),sin(31(1−𝑚))(√3−sin(𝜋31(1−𝑚𝜋))),if𝑚≡1(mod3),sin(31(1+𝑚))(√3−sin(𝜋31(1+𝑚))),if𝑚≡2(mod3). (1.9) Directly, one can see that √32>𝑏_𝑚≥0.86184088⋯,if(𝑚,3)=1,0.85559967⋯,if(𝑚,3)=3, (1.10) and thus, 𝜆_𝑚<𝜆,2.73205080⋯<𝜇_𝑚≤2.76364572⋯,if(𝑚,3)=1,2.81215109⋯,if(𝑚,3)=3. (1.11) Below, we prove the following results.

Theorem 1.1. If (𝑚,3)=1, then |𝑆_𝑚(𝑥)|≤1+𝜇_𝑚𝑥^𝜆𝑚. (1.12)

Theorem 1.2 (Generalized Newman phenomena). If 𝑚>3 is a multiple of 3, then |𝑆_𝑚3(𝑥)−𝑚𝑆₃(𝑥)|≤1+𝜇_𝑚𝑥^𝜆𝑚. (1.13)

Using Theorem 1.2 and (1.5), one can estimate 𝑥₀(𝑚) in (1.6). For example, one can prove that 𝑥₀(21)<𝑒⁹⁰⁹.

2. Explicit Formula for 𝑆_𝑚(𝑁)

We have 𝑆_𝑚(𝑁)=^𝑁−1_{𝑛=0,𝑚|𝑛}(−1)^𝑠(𝑛)=1𝑚^𝑚−1_𝑡=0^𝑁−1_𝑛=0(−1)^𝑠(𝑛)𝑒^{2𝜋𝑖(𝑛𝑡/𝑚)}=1𝑚^𝑚−1_𝑡=0^𝑁−1_𝑛=0𝑒^{2𝜋𝑖((𝑡/𝑚)𝑛+(1/2)𝑠(𝑛))}. (2.1) Note that the interior sum has the form 𝐹_𝛼(𝑁)=^𝑁−1_𝑛=0𝑒^{2𝜋𝑖(𝛼𝑛+(1/2)𝑠(𝑛))},0≤𝛼<1. (2.2)

Lemma 2.1. If 𝑁=2^𝜈0+2^𝜈1+⋯+2^𝜈𝑟,𝜈₀>𝜈₁>⋯>𝜈_𝑟≥0, then 𝐹_𝛼(𝑁)=^𝑟_ℎ=0𝑒^{2𝜋𝑖(𝛼∑ℎ−1𝑗=02𝜈𝑗𝜈+ℎ/2)ℎ−1}_𝑘=0(1+𝑒^{2𝜋𝑖(𝛼2𝑘+1/2)}), (2.3) where as usual ∑⁻¹_𝑗=0=0,∏⁻¹_𝑘=0=1.

Proof. Let 𝑟=0, then by (2.2), 𝐹_𝛼(𝑁)=^𝑁−1_𝑛=0(−1)^𝑠(𝑛)𝑒^{2𝜋𝑖𝛼𝑛}=1−^𝜈0−1_𝑗=0𝑒^{2𝜋𝑖𝛼2𝑗}+_{0≤𝑗1<𝑗2≤𝜈0−1}𝑒^{2𝜋𝑖𝛼(2𝑗1+2𝑗2)}=−⋯_{𝜈0−1𝑘=0}(1−𝑒^{2𝜋𝑖𝛼2𝑘}), (2.4) which corresponds to (2.3) for 𝑟=0.
Assuming that (2.3) is valid for every 𝑁 with 𝑠(𝑁)=𝑟+1, let us consider 𝑁₁=2^𝜈𝑟𝑎+2^𝜈𝑟+1 where 𝑎 is odd, 𝑠(𝑎)=𝑟+1, and 𝜈_𝑟+1<𝜈_𝑟. Let 𝑁=2^𝜈𝑟𝑎=2^𝜈0+⋯+2^𝜈𝑟;𝑁₁=2^𝜈0+⋯+2^𝜈𝑟+2^𝜈𝑟+1. (2.5) Notice that for 𝑛∈[0,2^𝜈𝑟+1), we have 𝑠(𝑁+𝑛)=𝑠(𝑁)+𝑠(𝑛). (2.6) Therefore, 𝐹_𝛼(𝑁₁)=𝐹_𝛼(𝑁)+^𝑁1−1_𝑛=𝑁𝑒^{2𝜋𝑖(𝛼𝑛+(1/2)𝑠(𝑛))}=𝐹_𝛼(𝑁)+^{2𝜈𝑟+1−1}_𝑛=0𝑒^{2𝜋𝑖(𝛼𝑛+𝛼𝑁+(1/2)(𝑠(𝑁)+𝑠(𝑛)))}=𝐹_𝛼(𝑁)+𝑒^{22𝜋𝑖(𝛼𝑁+(1/2)𝑠(𝑁))𝜈𝑟+1−1}_𝑛=0𝑒^{2𝜋𝑖(𝛼𝑛+(1/2)𝑠(𝑛))}. (2.7) Thus, by (2.3) and (2.4), 𝐹_𝛼(𝑁₁)=^𝑟_ℎ=0𝑒^{2𝜋𝑖(𝛼∑ℎ−1𝑗=02𝜈𝑗𝜈+ℎ/2)ℎ−1}_𝑘=0(1+𝑒^{2𝜋𝑖(𝛼2𝑘+1/2)}+𝑒^{2𝜋𝑖(𝛼∑𝑟𝑗=02𝜈𝑗𝜈+(𝑟+1)/2)𝑟+1−1}_𝑘=0(1+𝑒^{2𝜋𝑖(𝛼2𝑘+1/2)})=^𝑟+1_ℎ=0𝑒^{2𝜋𝑖(𝛼∑ℎ−1𝑗=02𝜈𝑗𝜈+ℎ/2)ℎ−1}_𝑘−0(1+𝑒^{2𝜋𝑖(𝛼2𝑘+1/2)}). (2.8)

Formulas (2.1)–(2.3) give an explicit expression for 𝑆_𝑚(𝑁) as a linear combination of the products of the form ^𝜈ℎ−1_𝑘=0(1+𝑒^{2𝜋𝑖(𝛼2𝑘+1/2)}𝑡),𝛼=𝑚,0≤𝑡≤𝑚−1. (2.9)

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) 𝑟≤𝜈₀=⌊ln𝑁ln2⌋. (3.1) By Lemma 2.1, we have |𝐹_𝛼(𝑁)|≤_{𝜈ℎ=𝜈0,𝜈1,…,𝜈𝑟}|^𝜈ℎ_𝑘=1(1+𝑒^{2𝜋𝑖(𝛼2𝑘−1+1/2)}≤)|_𝜈0ℎ=0|^ℎ_𝑘=1(1+𝑒^{2𝜋𝑖(𝛼2𝑘−1+1/2)})|. (3.2) Furthermore, 1+𝑒^{2𝜋𝑖(2𝑘−1𝛼+1/2)}=2sin(2^𝑘−1𝛼𝜋)(sin(2^𝑘−1𝛼𝜋)−𝑖cos(2^𝑘−1𝛼𝜋)) (3.3) and, therefore, |1+𝑒^{2𝜋𝑖(2𝑘−1𝛼+1/2)}|≤2|sin(2^𝑘−1𝛼𝜋)|. (3.4) According to (3.2), let us estimate the product ^ℎ_𝑘=1(2|sin(2^𝑘−1𝛼𝜋)|)=2^ℎℎ_𝑘=1|sin(2^𝑘−1𝛼𝜋)|, (3.5) where by (2.1), 𝑡𝛼=𝑚,0≤𝑡≤𝑚−1. (3.6) Repeating arguments of [1], put |sin(2^𝑘−1𝛼𝜋)|=𝑡_𝑘. (3.7) Considering the function 𝜌(𝑥)=2𝑥√1−𝑥²,0≤𝑥≤1, (3.8) we have 𝑡_𝑘=2𝑡_𝑘−11−𝑡²_𝑘−1=𝜌(𝑡_𝑘−1). (3.9) Note that𝜌^′(𝑥)=2(√1−𝑥²−𝑥²√1−𝑥²)≤−1 (3.10) for 𝑥₀≤𝑥≤1, where 𝑥₀=√32 (3.11) is the only positive root of the equation 𝜌(𝑥)=𝑥.

Show that either 𝑡_𝑘𝜋≤sin(𝑚⌊𝑚3𝜋⌋)=sin(𝑚⌈2𝑚3⌉)=𝑔_𝑚<√32 (3.12) or, simultaneously, 𝑡_𝑘>𝑔_𝑚, and 𝑡_𝑘𝑡_𝑘+1≤max_{0≤𝑙≤𝑚−1}(|sin𝑙𝜋𝑚|(√3−|sin𝑙𝜋𝑚=𝜋|))(sin(𝑚⌊𝑚3⌋))(√3−sin(𝜋𝑚⌊𝑚𝜋3⌋)),if𝑚≡1(mod3)(sin(𝑚⌈𝑚3⌉))(√3−sin(𝜋𝑚⌈𝑚3⌉)),if𝑚≡2(mod3)=ℎ_𝑚<34. (3.13) Indeed, let for a fixed values of 𝑡∈[0,𝑚−1] and 𝑘∈[1,𝑛], 𝑡2^𝑘−1≡𝑙(mod𝑚),0≤𝑙≤𝑚−1. (3.14) Then, 𝑡_𝑘=|sin𝑙𝜋𝑚|. (3.15) Now, distinguish two cases: (1) 𝑡_𝑘≤√3/2,(2)𝑡_𝑘>√3/2.

In case (1), 𝑡_𝑘=√32⇆𝑙𝜋𝑚=𝑟𝜋3,(𝑟,3)=1, (3.16) and since 0≤𝑙≤𝑚−1, then 𝑚=3𝑙𝑟,𝑟=1,2. (3.17) Because of the condition (𝑚,3)=1, we have 𝑡_𝑘<√3/2.

Thus, in (3.15), 𝑚𝑙∈[0,⌊3⌋]∪[⌈2𝑚3⌉,𝑚], (3.18) and (3.12) follows.

In case (2), let 𝑡_𝑘>√3/2=𝑥₀. For 𝜀>0, put 𝑡1+𝜀=^𝑘𝑥₀=2√3|sin(𝜋2^𝑘−1𝛼)| (3.19) such that 21−𝜀=2−√3|sin(𝜋2^𝑘−1𝛼)|,1−𝜀²=43|sin(𝜋2^𝑘−1𝛼)|(√3−|sin(𝜋2^𝑘−1𝛼)|). (3.20)(3.21) By (3.9) and (3.19), we have 𝑡_𝑘+1=𝜌(𝑡_𝑘)=𝜌((1+𝜀)𝑥₀)=𝜌(𝑥₀)+𝜀𝑥₀𝜌^′(𝑐), (3.22) where 𝑐∈(𝑥₀,(1+𝜀)𝑥₀).

Thus, according to (3.10) and taking into account that 𝜌(𝑥₀)=𝑥₀, we find 𝑡_𝑘+1≤𝑥₀(1+𝜀), (3.23) while by (3.19) 𝑡_𝑘=𝑥₀(1+𝜀). (3.24) Now, in view of (3.21) and (3.11), 𝑡_𝑘𝑡_𝑘+1≤|sin𝜋2^𝑘−1𝛼|(√3−|sin(𝜋2^𝑘−1𝛼)|), (3.25) and according to (3.14), (3.15), we obtain that 𝑡_𝑘𝑡_𝑘+1≤ℎ_𝑚, (3.26) where ℎ_𝑚 is defined by (3.13).

Notice that from simple arguments and according to (1.9), 𝑔_𝑚≤ℎ_𝑚=𝑏_𝑚. (3.27)

Therefore, ^ℎ_𝑘=1|sin(𝜋2^𝑘−1𝛼)|≤(𝑏_𝑚^⌊ℎ/2⌋)²≤𝑏_𝑚^ℎ−1. (3.28) Now, by (3.2)-(3.4), for 𝛼=𝑡/𝑚,𝑡=0,1,…,𝑚−1, we have |𝐹_𝑡/𝑚(𝑁)|≤^𝜈0_ℎ=0|^ℎ_𝑘=1(1+𝑒^{2𝜋𝑖(𝛼2𝑘−1+1/2)}≤)|_𝜈0ℎ=02^ℎℎ_𝑘=1|sin(2^𝑘−1𝛼𝜋)|≤1+2^𝜈0_ℎ=1(2𝑏_𝑚)^ℎ−1≤1+2(2𝑏_𝑚)^𝜈02𝑏_𝑚.−1 (3.29)

Note that, according to (1.7) and (3.1), (2𝑏_𝑚)^𝜈0=2^{𝜆𝑚𝜈0}≤2^{𝜆𝑚log2𝑁}=𝑁^𝜆𝑚. (3.30) Thus, by (1.8) |𝐹_𝑡/𝑚2(𝑁)|≤1+2𝑏_𝑚𝑁−1^𝜆𝑚=1+𝜇_𝑚𝑁^𝜆𝑚. (3.31) Thus, the theorem follows from (2.1).

4. Proof of Theorem 1.2

Select in (2.1) the summands which correspond to 𝑡=0,𝑚/3,2𝑚/3.

We have 𝑚𝑆_𝑚(𝑁)=^𝑁−1_𝑛=0(𝑒^{𝜋𝑖𝑠(𝑛)}+𝑒^{2𝜋𝑖(𝑛/3+(1/2)𝑠(𝑛))}+𝑒^{2𝜋𝑖(2𝑛/3+(1/2)𝑠(𝑛))})+^𝑚−1_{𝑡=1,𝑡≠𝑚/3,2𝑚/3}^𝑁−1_𝑛=0𝑒^{2𝜋𝑖((𝑡/𝑚)𝑛+(1/2)𝑠(𝑛))}. (4.1)

Since the chosen summands do not depend on 𝑚 and, for 𝑚=3, the latter sum is empty, then we find 𝑚𝑆_𝑚(𝑁)=3𝑆₃(𝑁)+^𝑚−1_{𝑡=1,𝑡≠𝑚/3,2𝑚/3}^𝑁−1_𝑛=0𝑒^{2𝜋𝑖((𝑡/𝑚)𝑛+(1/2)𝑠(𝑛))}. (4.2)

Further, the last double sum is estimated by the same way as in Section 3 such that |𝑆_𝑚3(𝑁)−𝑚𝑆₃(𝑁)|≤𝜇_𝑚𝑁^𝜆𝑚. (4.3)

Remark 4.1. Notice that from elementary arguments it follows that if 𝑚≥5 is a multiple of 3, then 𝜋(sin𝑚⌊𝑚−13⌋)(√3−sin𝜋𝑚⌊𝑚−13𝜋⌋)≤(sin𝑚⌈𝑚+13⌉)(√3−sin𝜋𝑚⌈𝑚+13⌉). (4.4) The latter expression is the value of 𝑏²_𝑚 in this case (see (1.9)).

Example 4.2. Let us find some 𝑥₀ such that 𝑆₂₁(𝑥)>0 for 𝑥≥𝑥₀.
Supposing that 𝑥 is multiple of 3 and using (1.4), we obtain that 𝑆₃2(𝑥)≥3^𝜆+1/2𝑥^𝜆. (4.5) Therefore, putting 𝑚=21 in Theorem 1.2, we have 𝑆₂₁1(𝑥)≥7𝑆₃(𝑥)−𝜇₂₁𝑥^𝜆212−1≥7⋅3^𝜆+1/2𝑥^𝜆−𝜇₂₁𝑥^𝜆21−1. (4.6) Now, calculating 𝜆 and 𝜆_𝑚 by (1.2) and (1.8), we find a required 𝑥₀: 𝑥₀=(3.5⋅3^𝜆+1/2𝜇₂₁)^{1/(𝜆−𝜆21)}=𝑒^908.379.... (4.7)

Corollary 4.3. For 𝑚 which is not a multiple of 3, denote 𝑈_𝑚(𝑥) the set of the positive integers not exceeding 𝑥 which are multiples of 𝑚 and not multiples of 3. Then, _{𝑛∈𝑈𝑚(𝑥)}(−1)^𝑠(𝑛)1=−𝑚𝑆₃(𝑥)+𝑂(𝑥^𝜆𝑚). (4.8)

In particular, for sufficiently large 𝑥, we have _{𝑛∈𝑈𝑚(𝑥)}(−1)^𝑠(𝑛)<0. (4.9)

Proof. Since |𝑈_𝑚(𝑥)|=𝑆_𝑚(𝑥)−𝑆_3𝑚(𝑥), (4.10) 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 𝑛∈ℕ,𝑛≠5,6, _𝑝≤𝑛(−1)^𝑠(𝑝)≤0, (5.1) where the summing is over all primes not exceeding 𝑛.

More precisely, by the observations, ∑_𝑝≤𝑛(−1)^𝑠(𝑝)<0 beginning with 𝑛=31. Moreover, the following conjecture holds.

Conjecture 5.2. lim_𝑛→∞ln(−∑_𝑝≤𝑛(−1)^𝑠(𝑝))=ln𝑛ln3ln4. (5.2)

A heuristic proof of Conjecture 5.2 was given in [8]. For a prime 𝑝, denote 𝑉_𝑝(𝑥) the set of positive integers not exceeding 𝑥 for which 𝑝 is the least prime divisor. Show that the correctness of Conjectures 5.1 (for 𝑛≥𝑛₀) follows from the following very plausible statement, especially in view of the above estimates.

Conjecture 5.3. For sufficiently large 𝑛, we have |_{√5≤𝑝≤𝑛𝑗∈𝑉𝑝(𝑛),𝑗>𝑝}(−1)^𝑠(𝑗)|<_{𝑗∈𝑉3(𝑛)}(−1)^𝑠(𝑗)=𝑆₃(𝑛)−𝑆₆(𝑛). (5.3)

Indeed, in the “worst case” (really is not satisfied), in which for all 𝑛≥𝑝²_{𝑗∈𝑉𝑝(𝑛),𝑗>𝑝}(−1)^𝑠(𝑗)<0,𝑝≥5, (5.4) we have a decreasing but positive sequence of sums: _{𝑗∈𝑉3(𝑛),𝑗>3}(−1)^𝑠(𝑗),_{𝑗∈𝑉3(𝑛),𝑗>3}(−1)^𝑠(𝑗)+_{𝑗∈𝑉5(𝑛),𝑗>5}(−1)^𝑠(𝑗),…,_{𝑗∈𝑉3(𝑛),𝑗>3}(−1)^𝑠(𝑗)+_{√5≤𝑝<𝑛𝑗∈𝑉𝑝(𝑛),𝑗>𝑝}(−1)^𝑠(𝑗)>0. (5.5) Hence, the “balance condition” for odd numbers [8] |_{𝑗≤𝑛,𝑗isodd}(−1)^𝑠(𝑗)|≤1 (5.6) must be ensured permanently by the excess of the odious primes. This explains Conjecture 5.1.

It is very interesting that for some primes 𝑝 the inequality (5.4), indeed, is satisfied for all 𝑛≥𝑝² . Such primes we call “resonance primes." Our numerous observations show that all resonance primes not exceeding 1000 are 11,19,41,67,107,173,179,181,307,313,421,431,433,587,601,631,641,647,727,787. (5.7)

In conclusion, note that for 𝑝≥3, we have lim_𝑛→∞|𝑉_𝑝(𝑛)|𝑛=1𝑝_{2≤𝑞<𝑝}1(1−𝑞) (5.8) such that lim_𝑛→∞(_𝑝≥3|𝑉_𝑝(𝑛)|𝑛1)=2. (5.9) Thus, using Theorems 1.1, 1.2 in the form 𝑆_𝑚(𝑛)=𝑜(𝑆₃3(𝑛)),(𝑚,3)=1,𝑚𝑆₃(𝑛)(1+𝑜(1)),3∣𝑚, (5.10) and inclusion-exclusion for 𝑝≥5, we find _{𝑗∈𝑉𝑝(𝑛)}(−1)^𝜎(𝑗)3=−3𝑝_{2≤𝑞<𝑝,𝑞≠3}1(1−𝑞)𝑆₃3(𝑛)(1+𝑜(1))=−2𝑝_{2≤𝑞<𝑝}1(1−𝑞)𝑆₃(𝑛)(1+𝑜(1)). (5.11)

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 𝑝≥5 and sufficiently large 𝑛≥𝑛_𝑝, we have _{𝑗∈𝑉𝑝(𝑛)}(−1)^𝑠(𝑗)<0 (5.12) and, moreover, lim_𝑛→∞ln(−∑_{𝑗∈𝑉𝑝(𝑛)}(−1)^𝑠(𝑗))=ln𝑛ln3ln4. (5.13)