Results using a certain kind of identity

Question

Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the use of a certain identity. To state it, let (for simplicity) $f$ be a completely multiplicative arithmetic function and let $$ \mathcal{M}(x) = \sum_{n\leq x} f(n). $$ Then the identity states $$ \mathcal{M}(x) \log x = \int_{1}^{x} \frac{\mathcal{M}(t)}{t}\ dt + \sum_{n\leq x} \Lambda(n)f(n) \mathcal{M}\left(\frac{x}{n}\right), $$ which is proved by summing $\sum_{n\leq x} f(n)\log n$ in two different ways. Here I'm really thinking of $f$ as the indicator function of some multiplicative property, such as

1. $f(n) = 1$ if the largest prime factor of $n$ is at most $y$, 0 otherwise;
2. $f(n) = 1$ if $(n,q)=1$, 0 otherwise.

Of course, the identity can be modified for certain properties that are not completely multiplicative (e.g. a congruence condition like $n\equiv a\ \text{mod}\ q$).

Hildebrand [1], proving an asymptotic for the number of $y$-smooth numbers up to $x$, gives a particularly elegant recursive argument which relies fundamentally on the above identity. Hildebrand's paper includes a few references that use this kind of identity. As well, Granville [2] uses this kind of identity in a few of his papers on smooth numbers in arithmetic progressions.

My question is, what other results in the literature use this kind of identity? By "this kind of identity", I mean specifically an identity derived by summing $\sum_{n\leq x} f(n)\log n$ using (1) partial summation and (2) using $\log n = \sum_{d\mid n}\Lambda(n)$.

I realize this is a bit of an open-ended question, but I find this idea elegant and would like to know of other places where it is applied. Any references are most appreciated.

References to the papers I've looked at:

[1] Hildebrand, "On the Number of Positive Integers $\leq x$ and Free of Prime Factors $> y$

[2] Granville, "Integers, without large prime factors, in arithmetic progressions, I and II"

Answer 1

Hildebrand, in the paper you cite, discusses briefly its origins (last paragraph of the introduction).

This identity seems to have first appeared in Delange's paper in 1961, titled "Sur les fonctions arithmétiques multiplicatives" (Ann. Sci. École Norm. Sup. (3) 78 1961 273–304). It has been the starting point of many investigations in multiplicative number theory, old and new. It allows one to study the mean value of a function through its mean value on primes, and to reduce questions about multiplicative functions to question on integral equations.

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I'll start by summarizing the general formula. Given any multiplicative function $f$, subject only to $f(1)=1$, one can define an arithmetic function $\Lambda_f$ uniquely through the relation $$f * \Lambda_f = f \cdot \log.$$ This relation can also be encoded in terms of (formal) Dirichler series: $$ -\frac{F'(s)}{F(s)} = \sum_{n \ge 1} \frac{\Lambda_f(n)}{n^s} \qquad \left(F(s) =\sum_{n \ge 1} \frac{f(n)}{n^s}\right).$$ This later definition also shows $\Lambda_f$ is supported on prime powers. For any prime $p$ we have the recursive relation $$\sum_{j=0}^{k-1} f(p^j)\Lambda_f(p^{k-j}) = f(p^k)\log(p^k), \qquad k \ge 1.$$ If $f$ is completely multiplicative, as in your question, one can verify $\Lambda_f(n)=\Lambda(n)f(n)$.

On the one hand, $$\sum_{n \le x} f(n)\log n = \sum_{n \le x} (f*\Lambda_f)(n) = \sum_{p^m \le x} \Lambda_f(p^m) \mathcal{M}(x/p^m)$$ where I used your notation $\mathcal{M}(x) = \sum_{n \le x} f(n)$. On the other hand, Abel summation shows $$\sum_{n \le x} f(n)\log n=\mathcal{M}(x) \log x - \int_{1}^{x} \frac{\mathcal{M}(t)}{t}dt.$$ Equating these two identities gives $$(\star)\, \mathcal{M}(x)\log x = \int_{1}^{x} \frac{\mathcal{M}(t)}{t}dt + \sum_{n \le x} \Lambda_f(n) \mathcal{M}\left(\frac{x}{n}\right).$$

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In the aforementioned paper of Delange, you will find the definition of $\Lambda_f$ in p. 277, equation (2) (where he uses the notation $c_j^{(p)}(f)$ for $\Lambda_f(p^j)/\log p$). In that paper, Delange considers $1$-bounded (possibly complex-valued) multiplicative functions, and proves in particular that if $\sum_{p} (1-f(p))/p$ converges then $f$ has an (explicit) mean value, and conversely, if $f$ has a non-zero mean value then $\sum_{p} (1-f(p))/p$ converges and $f(2^k)\neq -1$ for some $k\ge 1$.

The definition of $\Lambda_f$ is already useful in itself. Halász has a famous inequality holding for $1$-bounded multiplicative functions. It turns out that the correct framework for generalizing it to divisor-bounded multiplicative functions involves $\Lambda_f$: let $C(\kappa)$ be the set of functions with $|\Lambda_f(n)|\le \kappa \Lambda(n)$. Granville, Harper and Soundararajan, in "A new proof of Halász’s theorem, and its consequences" (Compos. Math. 155, No. 1, 126-163 (2019)), among other things, generalize the inequality to this class. See also their related paper "Mean values of multiplicative functions over function fields" (Res. Number Theory 1, Paper No. 25, 18 p. (2015)).

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As a general principle, $(\star)$ allows one to study the mean value of a function through its mean value on primes. Letting $$\sigma_f(u) := y^{-u} \sum_{n \le y^u} f(n)$$ and $$\chi_f(u):= \left(\sum_{n \le y^u} \Lambda(n)\right)^{-1} \sum_{n \le y^u} \Lambda_f(n) \approx y^{-u} \sum_{p \le y^u} f(p)\log p,$$ then $(\star)$ can be used to show that $\sigma_f$ and $\chi_f$ approximately satisfy the integral equation $$(\star \star) \, u\sigma(u) = \sigma*\chi(u):=\int_{0}^{u}\sigma(u-t)\chi(t)dt.$$ This idea has been used successfully by Wirsing in "Das asymptotische Verhalten von Summen über multiplikative Funktionen. II" (Acta Math. Acad. Sci. Hung. 18, 411-467 (1967)), where he proves (among other things) an asymptotic formula for $\sum_{n \le x}f(n)$ under very general conditions.

The connection between $(\star)$ and $(\star \star)$ has been used many times since. See especially "The spectrum of multiplicative functions" by Granville and Soundararajan (Ann. Math. (2) 153, No. 2, 407-470 (2001)), where in Proposition 1 and its converse they show in rather wide generality a correspondence between $(\sigma_f,\chi_f)$ and the solutions $(\sigma,\chi)$ to $(\star \star)$ (in both directions). (See their result for accurate formulations.)

Another application of $(\star \star)$ appears in "The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture" by Tao (Algebra Number Theory 9, No. 4, 1005-1034 (2015)).