# On the Correlation of Multiplicative and the Sum of Additive Arithmetic Functions

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*P. D. T. A. Elliott*

This work applies stability properties of the dual of a certain arithmetic operator to study the correlation of multiplicative arithmetic functions. The literature of number theory contains very little concerning such correlations despite their direct connection with the problem of prime pairs and Goldbach's conjecture concerning the representation of even integers as the sum of two primes. Elliott aims for a result of wide uniformity under very weak hypotheses. The uniformity obtained here enables a comprehensive investigation of the value distribution of sums of additive arithmetic functions on distinct arithmetic progressions. The underlying argument, which Elliott calls the method of the stable dual, has received no unified account in the literature. A short overview of the method, with historical remarks, is presented. The principal results presented here are all new and currently beyond the reach of any other method.

#### Table of Contents

# Table of Contents

## On the Correlation of Multiplicative and the Sum of Additive Arithmetic Functions

- Contents v6 free
- Introduction vii8 free
- 1. Correlations of Multiplicative Functions 110 free
- 2. The Method of the Stable Dual (1): Deriving the Approximate Functional Equations 1625
- 3. The Method of the Stable Dual (2): Solving the Approximate Functional Equations 2231
- 4. Orthogonality of Operators and the Present Application of the Method to the Study of Correlations 3948
- 5. Correlations: Main Lemma in the Proof of Theorem 1.1 4352
- 6. Correlations: Auxiliary Lemmas for the Proof of Theorem 1.1 4655
- 7. Correlations: An Approximate Functional Equation and Its Solution 5362
- 8. Correlations: Proof of the Main Lemma (completion) 5968
- 9. Correlations: Proof of Theorem 1.1 6271
- 10. Sums of Additive Functions 6675
- 11. Afterwords: Further Possibilities 8291
- References 8493