[Paper Review] On Popov's formula involving the Von Mangoldt function

Alexander E. Patkowski|arXiv (Cornell University)|May 15, 2017

Analytic Number Theory Research2 references3 citations

TL;DR

This paper provides the first known proof of Popov's 1943 formula involving the von Mangoldt function and fractional parts, generalizing it to arbitrary real $ r \geq 1 $ using Mellin transforms and contour integration. The key contribution is a new explicit formula relating sums over $ \Lambda(n) $ to non-trivial zeros $ \rho $ of the Riemann zeta function and trivial zeros, with analogous results for the Möbius function.

ABSTRACT

We offer a generalization of a formula of Popov involving the Von Mangoldt function. Some commentary on its relation to other results in analytic number theory is mentioned as well as an analogue involving the m$\ddot{o}$bius function.

Motivation & Objective

• To provide a rigorous proof of Popov's unproven 1943 formula involving the von Mangoldt function and fractional parts.
• To generalize Popov's formula to arbitrary real $ r \geq 1 $, extending its applicability.
• To establish an analogous formula for the Möbius function $ \mu(n) $, linking it to zeta function zeros.
• To clarify the connection between arithmetic sums with fractional parts and the non-trivial zeros of the Riemann zeta function.
• To demonstrate the utility of Mellin transform and contour integration techniques in analytic number theory for such sums.

Proposed method

• Use of the Mellin transform of $ \{t\} - \frac{1}{2} $, known to yield $ \frac{s+1}{2s(s-1)} - \frac{\zeta(s)}{s} $ for $ \Re(s) > 1 $.
• Application of Mellin inversion to express $ \frac{1}{2}(\{u\}^2 - \{u\}) $ as a contour integral involving $ \zeta(s) $ and $ s $.
• Substitution $ u = \frac{n}{x} $ and summation over $ n > x $, leading to an integral involving $ \zeta'(r-s)/\zeta(r-s) $.
• Change of variables $ s \to 1-s $ to re-express the integral in terms of $ \zeta(s+r-1) $, revealing poles at $ s=1 $, $ s=2-r $, $ s=\rho+1-r $, and $ s=-2k-r+1 $.
• Computation of residues at these poles to derive the main formula, with the remaining integral vanishing due to absolute convergence.
• Adaptation of the method to the Möbius function case, using the functional equation of $ \zeta(s) $ to evaluate residues at trivial zeros.

Experimental results

Research questions

• RQ1 What is the correct generalization of Popov's formula for $ r \geq 1 $, and how can it be rigorously proven?
• RQ2 How do the non-trivial zeros $ \rho $ of the Riemann zeta function emerge in sums involving the von Mangoldt function and fractional parts?
• RQ3 Can the same method be adapted to derive a formula for the Möbius function $ \mu(n) $, and what role do the trivial zeros play?
• RQ4 What is the significance of the constant term $ h_r(x) $, and how does it relate to the poles at $ s=1 $ and $ s=2-r $?
• RQ5 How does the Mellin transform technique facilitate the derivation of such explicit formulas in analytic number theory?

Key findings

• The generalized formula (1.2) holds for all real $ r \geq 1 $, with the sum over $ \Lambda(n) $ expressed via residues at $ s=1 $, $ s=2-r $, $ s=\rho+1-r $, and $ s=-2k-r+1 $.
• For $ r=1 $, the formula reduces to Popov's original result, with the constant term $ h_1(x) = \frac{2 - \log(2\pi)}{2x} $.
• For $ r > 1 $, the constant term is $ h_r(x) = \left( \frac{r}{2(2-r)(1-r)} - \frac{\zeta(r-1)}{1-r} \right) \frac{x^{-r}}{r} $, arising from the pole at $ s=2-r $.
• The sum over non-trivial zeros $ \rho $ appears as $ \sum_{\rho} \left( \frac{r+1-\rho}{2(\rho+1-r)(\rho-r)} - \frac{\zeta(r-\rho)}{r-\rho} \right) \frac{x^{\rho-r-1}}{r+1-\rho} $.
• The sum over trivial zeros contributes terms involving $ \zeta(2k+r) $, with coefficients derived from the functional equation of $ \zeta(s) $.
• An analogous formula is derived for the Möbius function $ \mu(n) $, where the residue at $ s=1 $ vanishes and the trivial zero residues involve $ \frac{1}{\zeta'(-2k)} $, yielding a new explicit formula involving $ \pi^{2k} $, $ \zeta(2k+1) $, and factorials.