[Paper Review] Finite and infinite Euler products of Ramanujan expansions

Giovanni Coppola|arXiv (Cornell University)|Oct 31, 2019

Analytic Number Theory Research10 references4 citations

TL;DR

This paper establishes a finite Euler product factorization for Ramanujan expansions of arithmetic functions when the Ramanujan coefficients are multiplicative and pointwise convergence holds. The key result shows that such expansions can be decomposed into a finite product over a chosen finite set of primes and an infinite tail, with the finite part depending on the p-adic valuation of the argument, providing a structural decomposition with applications to the null function and multiplicative coefficients.

ABSTRACT

All the $F:$N$ ightarrow $C having Ramanujan expansion $F(a)=\sum_{q=1}^{\infty}G(q)c_q(a)$ (here $c_q(a)$ is the Ramanujan sum) pointwise converging in $a\in $N, with $G:$N$ ightarrow $C a multiplicative function, may be factored into two Ramanujan expansions, one of which is a finite Euler product : details in our Main Theorem. This is a general result, with unexpected and useful consequences, esp., for the Ramanujan expansion of null-function, say 0. The Main Theorem doesn't require other analytic assumptions, as pointwise convergence suffices; this depends on a general property of Euler $p-$factors (the factors in Euler products) for the general term $G(q)c_q(a)$; namely, once fixed $a\in $N (and prime $p$), the $p-$Euler factor of $G(q)c_q(a)$ (involving all $p-$powers) has a finite number of non-vanishing terms (depending on $a$) : see our Main Lemma. In case we also add some other hypotheses, like the absolute convergence, we get more classical Euler products: the infinite ones. For the Ramanujan expansion of 0 this strong hypothesis makes the class of 0 Ramanujan coefficients much smaller; also excluding Ramanujan's $G(q)=1/q$ and Hardy's $G(q)=1/φ(q)$ ($φ$ is Euler's totient function). Our Main Theorem, instead, suffices to classify all the multiplicative Ramanujan coefficients for 0, so we also announce and (partially) prove this Classification.

Motivation & Objective

• To establish a general factorization of Ramanujan expansions into finite and infinite Euler products under multiplicative coefficients and pointwise convergence.
• To analyze the structure of Ramanujan coefficients for the null function, particularly focusing on multiplicative functions.
• To classify multiplicative Ramanujan coefficients supported on smooth numbers and determine conditions under which they vanish identically.
• To provide a theoretical foundation for understanding why classical coefficients like Ramanujan’s G(q) = 1/q and Hardy’s G(q) = 1/φ(q) do not satisfy absolute convergence.

Proposed method

• Introduce a finite set of primes F and decompose the Ramanujan expansion into a finite Euler product over F and a complementary tail over integers coprime to F.
• Use the Main Lemma to show that for fixed a ∈ ℕ, the p-adic factor of G(q)c_q(a) has only finitely many non-zero terms, depending on v_p(a).
• Apply the multiplicative property of G to express the finite factor as a product over p ∈ F of sums up to v_p(a), involving differences G(p^K) - G(p^{K+1}).
• Establish convergence via pointwise convergence assumptions, avoiding stronger analytic conditions like absolute convergence.
• Use the structure of the Ramanujan sum c_q(a) and its relation to the Möbius function μ(q) when a = 1 to derive necessary conditions for vanishing expansions.
• Leverage the Eratosthenes transform and smooth number support to analyze coefficient support and derive conditions for the null function.

Experimental results

Research questions

• RQ1 Under what conditions can a Ramanujan expansion with multiplicative coefficients be factored into a finite Euler product and an infinite tail?
• RQ2 Why do classical Ramanujan coefficients like G(q) = 1/q and G(q) = 1/φ(q) fail to satisfy absolute convergence, and what does this imply for their representation?
• RQ3 What structural constraints must multiplicative Ramanujan coefficients satisfy to represent the null function identically?
• RQ4 How does the p-adic valuation v_p(a) influence the length and form of the finite Euler product in the expansion?
• RQ5 What is the role of the set of primes F(G) in classifying multiplicative Ramanujan coefficients of the null function?

Key findings

• The Main Theorem establishes that for any multiplicative coefficient function G and finite set of primes F, the Ramanujan expansion factors as a finite Euler product over F and an infinite tail over integers coprime to F, provided pointwise convergence holds.
• The finite Euler product is explicitly given by ∏_{p∈F} ∑_{K=0}^{v_p(a)} p^K (G(p^K) - G(p^{K+1})), showing dependence on the p-adic valuation of the argument a.
• For the null function, the condition ∑_{(r,F)=1} G(r)μ(r) = 0 is necessary and sufficient for the Ramanujan expansion to vanish identically, under pointwise convergence.
• The paper shows that Ramanujan’s G(q) = 1/q and Hardy’s G(q) = 1/φ(q) are not absolutely convergent, explaining why they lie outside the class of coefficients admitting infinite Euler products.
• The finite Ramanujan expansion over Q-smooth numbers is shown to be finite due to the dependence on v_p(a), even though the support set is infinite.
• The classification of multiplicative Ramanujan coefficients of the null function is completed by showing that only three cases arise: F(G) = ∅, F_0(G) ≠ ∅, or F(G) ≠ ∅ and F_0(G) = ∅, with the last case requiring the sum condition on μ(r) to vanish.