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Note on the Theory of Correlation Functions Note On The Theory Of Correlation And Autocor- relation of Arithmetic Functions N. A. Carella Abstract: The correlation functions of some arithmetic functions are investigated in this note. The upper bounds for the autocorrelation of the Mobius function of degrees two and three C C n x µ(n)µ(n + 1) x/(log x) , and n x µ(n)µ(n + 1)µ(n + 2) x/(log x) , where C > 0 is a constant,≤ respectively,≪ are computed using≤ elementary methods. These≪ results are summarized asP a 2-level autocorrelation function. Furthermore,P some asymptotic result for the autocorrelation of the vonMangoldt function of degrees two n x Λ(n)Λ(n +2t) x, where t Z, are computed using elementary methods. These results are summarized≤ as a 3-lev≫el autocorrelation∈ function. P Mathematics Subject Classifications: Primary 11N37, Secondary 11L40. Keywords: Autocorrelation function, Correlation function, Multiplicative function, Liouville function, Mobius function, vonMangoldt function. arXiv:1603.06758v8 [math.GM] 2 Mar 2019 Copyright 2019 ii Contents 1 Introduction 1 1.1 Autocorrelations Of Mobius Functions . ..... 1 1.2 Autocorrelations Of vonMangoldt Functions . ....... 2 2 Topics In Mobius Function 5 2.1 Representations of Liouville and Mobius Functions . ...... 5 2.2 Signs And Oscillations . 7 2.3 DyadicRepresentation............................... ... 8 2.4 Problems ......................................... 9 3 SomeAverageOrdersOfArithmeticFunctions 11 3.1 UnconditionalEstimates.............................. ... 11 3.2 Mean Value And Equidistribution . 13 3.3 ConditionalEstimates ................................ .. 14 3.4 DensitiesForSquarefreeIntegers . ....... 15 3.5 Subsets of Squarefree Integers of Zero Densities . ........... 16 3.6 Problems ......................................... 17 4 Mean Values of Arithmetic Functions 19 4.1 SomeDefinitions ..................................... 19 4.2 Some Results For Arithmetic Functions . .... 19 4.3 WirsingFormula ..................................... 23 4.4 Result In Comparative Multiplicative Functions . ..... 23 4.5 Extension To Arithmetic Progressions . ..... 24 5 CorrelationsofMobiusFunctionsOfDegreeTwo 25 5.1 MobiusAutocorrelationofDegreeTwo. ..... 25 5.2 Liouville Autocorrelation of Degree Two . ... 27 5.3 ConditionalUpperBounds ............................. .. 28 5.4 Correlation For Squarefree Integers of Degree Two . .......... 29 5.5 Comparison........................................ 30 5.6 Problems ......................................... 30 6 CorrelationofSomeBoundedMultiplicativeFunctions 33 6.1 Correlation of Some Bounded Multiplicative Functions . ..... 33 6.2 Independent Proof For Some Bounded Multiplicative Functions . ........ 35 6.3 Correlation Functions of Higher Degrees . ...... 35 7 PolynomialsIdentitiesForTheCorrelationFunctions 37 7.1 Quadratic Identity for the Autocorrelation of the Mobius Function......... 37 7.2 MomentsofMobiusFunction . .. .. .. .. .. .. .. .. .. .. .. .. .. 38 7.3 Analytic Expression for the Mobius Function . ..... 38 7.4 Analytic Expression for the Louiville Function . ..... 39 iii iv Contents 8 Autocorrelation of the vonMangoldt Function 41 8.1 Representations of the vonMangoldt Function . ........ 42 8.2 PrimeNumbersTheorems.............................. .. 42 8.3 Autocorrelation of the vonMangoldt Function . ....... 43 8.4 ElementaryProof................................... .. 44 8.5 AThreeLevelsAutocorrelation. .... 46 8.6 Fractional vonMangoldt Autocorrelation Function . ........ 46 8.7 Problems ......................................... 48 9 Correlations Functions Of Degree Three 51 9.1 UnconditionalEstimates.............................. ... 51 9.2 ConditionalUpperBounds ............................. .. 53 9.3 Comparison........................................ 53 9.4 Problems ......................................... 53 10 Correlations Functions Of Higher Degrees 55 10.1 UnconditionalEstimates. .... 55 10.2 ConditionalUpperBounds . ... 57 10.3Problems ......................................... 58 11 Some Arithmetic Correlation Functions 59 11.1 Divisors Correlation Functions . .... 59 11.2 Autocorrelation of the Sum of Divisors Function . ....... 59 11.3 Autocorrelation of the Euler Totient Function . ....... 60 11.4 Divisor And vonMangoldt Correlation Functions . ...... 60 11.5 Shifted vonMangoldt Correlation Function . ....... 60 11.6 CharactersCorrelationFunctions . ....... 61 12 Improved Correlation Result 63 12.1MainResults ....................................... 63 12.2Comparison....................................... 64 13SomeAverageOrdersOfArithmeticFunctions 65 13.1 Correlation Functions over Subsets of Integers . ........... 65 14 Exponential Sums Estimates 67 14.1SharperEstimate.................................. ... 67 14.2 ConditionalUpperBound .. .. .. .. .. .. .. .. .. .. .. .. ... 67 14.3Problems ......................................... 68 15 Twisted Arithmetic Sums 69 16 Fractional Functions 71 Chapter 1 Introduction Let f,g : N C be arithmetic functions, and let τ1, τ2,...τk N, and let k 1 be fixed integers. The k-degree−→ autocorrelation and correlation function of f and∈g are defined≥ by R(τ)= f(n + τ )f(n + τ ) f(n + τ ). (1.1) 1 2 ··· k n x X≤ and C(τ)= f(n + τ )g(n + τ )g(n + τ )g(n + τ ) f(n + τ )g(n + τ ) (1.2) 1 1 2 1 ··· k k n x X≤ respectively. Some combinations of the parameters and functions are easy to evaluate but other are not. In particular, if f(n)= µ(n), the case τ1 = τ2 = = τk, with k 2Z, is usually not dif- ficulty to estimate or calculate. But the case τ = τ for some···i = j is usually∈ a challenging problem. i 6 j 6 Trivially, the autocorrelation function has the upper bound R(τ) f k x, where | |≪k k f = max f(n) . (1.3) n N k k ∈ {| |} And a priori, a random sequence f(n),f(n + 1),f(n + 3),... is expected to have the upper bound R(τ) f k x1/2(log x)B , where B > 0 is a constant, see [16], [17], [[18], Theorem 2], and [2]. | |≪k k Some extreme cases such that R(τ) f k x, are demonstrated in [62]. This note deals with the correlation functions for some| arithmetic|≫k k functions. The main results are Theorem 1.1 for the Mobius function, Theorem 1.2 for the Liouville function, and Theorem 1.2 for the vonMangoldt function. 1.1 Autocorrelations Of Mobius Functions The study of the Mobius function and various forms of its autocorrelation is closely linked to the autocorrelations and distribution functions of binary sequences. The purpose is to improve the current theoretical results for the Mobius sequence µ(n): n 1 and the Liouville sequence λ(n): n 1 . Given a large number x 1, these{ results≥ imply} that the short sequences {µ(n),µ(n +≥ 1),...,µ} (n + r 1) and λ(n)≥, λ(n + 1),...,λ(n + r 1) of length r = [x] 1 are{ well distributed and have− 2-level} correlation{ functions. The theory− and} recent results for≥ the correlation functions of some binary sequences are investigated in [17], et alii. Theorem 1.1. Let C > 2 be a constant, and let k 1 be a small fixed integer. Then, for every large number x> 1, ≥ x µ(n + τ )µ(n + τ ) µ(n + τ )= O (1.4) 1 2 ··· k (log x)C n x X≤ for any fixed sequence of integers τ < τ < < τ . 1 2 ··· k 1 2 Chapter 1. Introduction This result improves the upper bound for the average order of the correlations of Mobius functions log log T 1 k µ(n + τ1)µ(n + τ2) µ(n + τk) k + T x , (1.5) ··· ≪ log T log1/3000 x 1 τi T x n 2x ≤X≤ ≤X≤ where 10 T x, see [28, p. 4], and [65]. Other related works are the special case ≤ ≤ µ(n)µ(r n)= O(x/(log x)B ) (1.6) − n r X≤ for almost every r > 1, and B > 0 constant, which is proved in [23], and the functions fields versions in [12], [14], and [66]. Specifically, there is an explicit upper bound n 1/2 2 n 1 µ(F + f1)µ(F + f2) µ(F + fk) 2knq − +3rn q − , (1.7) ··· ≤ F Mn X∈ where fi Fq[x ] are distinct fixed polynomials of degree deg(fi) < n, and Mn = F Fq[x]: deg(F )=∈n is the subset of polynomials of degree deg(f )= n, this appears in [14]. { ∈ } i Theorem 1.2. Let C > 2 be a constant, and let k 1 be a small fixed integer. Then, for every large number x> 1, ≥ x λ(n + τ )λ(n + τ ) λ(n + τ )= O (1.8) 1 2 ··· k (log x)C n x X≤ for any fixed sequence of integers τ < τ < < τ . 1 2 ··· k C The upper bound for simpler correlation functions n x µ(n)µ(n + 1) x/(log x) , in Theorem ≤ ≪ ?? in Section ??, and µ(n)µ(n + 1)µ(n + 2) x/(log x)C , in Theorem 9.1 in Section 9.1, are n x P considered first. The more≤ general proofs for Theorem≪ 1.1 and Theorem 1.2 appear in Section 10.1. P 1.2 Autocorrelations Of vonMangoldt Functions The correlation of the vonMangoldt function is a long standing problem in number theory. It is the foundation of many related conjectures concerning the primes pairs such as p and p+2k as p , → ∞ where k 1 is fixed, and the prime m-tuples p,p+a1,p+a2,...,p+am, where a1,a2,...,am is an admissible≥ finite sequence of relatively prime integers. The most recent progress in the theory of the correlation of the von Mangoldt function are the various results based on the series of papers authored by [37]. These authors employed an approximation ΛR(n) of the function Λ(n) to obtain the estimate k x Λ (n)Λ (n + k)= G(k)x + O + O R2 , (1.9) R R ϕ(k) (log 2R/k)C n x X≤ confer Theorem 8.3 for the precise details. The density is defined by the singular series p 1 1 G(h)=2 − 1 . (1.10) p 2 − (p 1)2 p h n>2 Y| − Y − Another result in [39, Theorem 1.3] has a claim for the average Hardy-Littlewood conjecture, it covers almost all shift of the autocorrelation, but no specific value. A synopsis of the statement is recorded below. x Λ(n)Λ(n + h)= G(h)x + O (1.11) (log x)C n x X≤ 1.2. Autocorrelations Of vonMangoldt Functions 3 C for all but O H/ log x values h with h h0 H, confer Theorem 8.4 for the precise details. The function fields version of these von Mangoldt| − | ≤ correlations results are proved in [57].
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