Euler Valuation Moment Comparison

Abstract

An explicit logarithmic comparison of full valuation and prime-support Euler moments.

For a prime p and a natural exponent a, Z_p(a) is the sum of p^(-j) for 0 <= j <= a. Define U_p(s) as (1-1/p) times the sum over a >= 0 of Z_p(a)^s p^(-a), and define W_p(s) as 1-1/p+(1/p)(1-1/p)^(-s). Real powers are used. U(s) and W(s) are the products of these local factors over every prime.

Theorem 1.1 (Convergence and logarithmic comparison).

$$\begin{array}{r}\forall s\in \mathbb{R},4\le s\Rightarrow \\ \forall pprime,Summable\left(a\mapsto Z_p{\left(a\right)}^s{p}^{-a}\right),1\le U_p\left(s\right)\le W_p\left(s\right)\\ Multipliable\left(p\mapsto U_p\left(s\right)\right),Multipliable\left(p\mapsto W_p\left(s\right)\right)\\ 0<U\left(s\right),0<W\left(s\right)\\ 0\le \log \left(W\left(s\right)\right)-\log \left(U\left(s\right)\right)\le \sqrt{s}(\log \left(s\right)+5)\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/EulerValuationMomentComparison.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The real moment order s is at least four. Every local valuation series converges, the local factors lie between one and W_p(s), and both full prime products converge to strictly positive real numbers. Multipliable denotes convergence of the net of finite products.

A single term near the critical exponent log(s)/log(p) controls the logarithmic loss for p <= sqrt(s). For larger primes the local loss is at most s/(p squared minus one), and the integer tail telescopes. Combining the two ranges gives the displayed bound.

The definition and large-moment asymptotic of W(s) are given in Andreas Weingartner, The distribution functions of sigma(n)/n and n/phi(n), II, arXiv:1011.4262v1, equation (5) and Lemma 5. The explicit comparison here is the derivation in the Fibonacci atomic relation volume; no originality claim is made.

References

• Truth anchor: D5/S3/Arith/Robin/EulerValuationMomentComparison.result
• Dependency: D5/S3/Arith/GoldenResource/ChainResponseSelectorGap
• Dependency: D5/S3/Arith/GoldenResource/PrefixDeficitKernel