Reference to a variant of Abel's summation formula

Question

Edit. A stronger version of the formula is true (details follow).

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that $\lambda_n \to \infty$ as $n \to \infty$, and $f$ a function $\mathbf R^+ \to \mathbf C$ that admits a derivative $f^\prime(x)$ at all but countably many points $x \in [\lambda_1, \infty[$. Then, for $x \ge \lambda_1$ we have $$\sum_{n:\lambda_n \le x} a_n f(\lambda_n) = A(x) f(x) - \int_{\lambda_1}^x A(t) f^\prime(t) dt,$$ where we take $A(t) := \sum_{n: \lambda_n \le t} a_n$ for all $t \in \bf R$ and the integral on the right-hand side is understood in the sense of Henstock and Kurzweil.

This is a (straightforward) generalization of Abel's summation formula in two ways: First, $f^\prime$ is not required to be continuous or to exist everywhere in $[\lambda_1, \infty[$, and second, we don't assume $\lambda = n$ for all $n$.

A weaker version of the above variant of Abel's summation formula is mentioned, e.g., in some lecture notes by Alessandro Zaccagnini (Theorem A.1.1, p. 175), where $f$ is actually continuously differentiable (not that this makes any serious difference from the point of view of the proof, if you are already familiar with Theorem 4.7 in Bartle's A Modern Theory of Integration).

Question. Do you know of a more standard source where I can find it?

I would like to cite it in something we are writing, but all the sources that I've so far checked (*) deal with the classical version where $\lambda_n = n$ for all $n$ and $f^\prime$ exists and is continuous everywhere in $[\lambda_1, \infty[$.

Of course, we could just say that the proof of the classical formula carries over almost verbatim, or include a sketchy proof of the result, but a reference would be better (at least in my head).

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(*) To wit, Theorem 4.2 in Apostol's Introduction to Analytic Number Theory, Theorem 1 in Chapter I.0 of Tenenbaum's Introduction to analytic and probabilistic number theory, Theorem A.2.3 in Pollack's Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, and Theorem A.4 in Nathanson's Additive Number Theory: The Classical Bases.

Answer 1

In the book

K. Chandrasekharan, Arithmetical functions, Grundlehren math. Wiss. 167, Springer, 1970.

(page 22) Abel's summation formula is given in the following form:

Let $0 \leqslant \lambda_1 \leqslant \lambda_2 \leqslant \ldots$ be a sequence of real numbers, such that $\lambda_n \rightarrow \infty$ as $n \rightarrow \infty$, and let $\left(a_n\right)$ be a sequence of complex numbers. Let $A(x)=\sum_{\lambda_n \leqslant x} a_n$, and $\varphi(x)$ a complex-valued function defined for $x \geqslant 0$. Then $$ \sum_{n=1}^k a_n \varphi\left(\lambda_n\right)=A\left(\lambda_k\right) \varphi\left(\lambda_k\right)-\sum_{n=1}^{k-1} A\left(\lambda_n\right)\left(\varphi\left(\lambda_{n+1}\right)-\varphi\left(\lambda_n\right)\right) . $$ If $\varphi$ has a continuous derivative $\varphi^{\prime}$ in $(0, \infty)$, and $x \geqslant \lambda_1$, then $$ \sum_{\lambda_n \leqslant x} a_n \varphi\left(\lambda_n\right)=A(x) \varphi(x)-\int_{\lambda_1}^x A(t) \varphi^{\prime}(t) d t. $$

The required generalization of Abel's summation follows from the first formula.

Answer 2

Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was assumed to exist and be continuous everywhere in $[\lambda_1, \infty[$: The result (with some extra) appears as Theorem 6 in Chapter VII of

K. Chandrasekharan, Introduction to Analytic Number Theory, Grundlehren math. Wiss. 148, Springer, 1968.

As I don't have much hope that a variant of the formula in the lines of the latest version of the OP has ever appeared in print (in spite of the fact that it's straightforwardly generalized from the variant in Chandrasekharan's book, once you know you have a suitable version of the fundamental theorem of calculus for the Henstock-Kurzweil integral), I'm going to accept this as an answer.