Refactorisation of the Dirichlet convolution

2022/06/10 by Ansar El Hassani, Hassani, Ansar El
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical and Theoretical Analysis #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.05371

openalex publication_date 2022/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a new way to factor the dirichlet convolution for completely multiplicative functions whitch led us to constructing a ring that arise from the operations involved in the factorisation. We will conclude by some identities that was found during this work. An application of the results gives us a generalisation of the following Hardy formula: ζ(x)² = ζ(2x)∑_m=1^+∞ \frac2^ω(m)m^x which is: |ζ(z)|² = ζ(2x)∑_m=1^+∞\frac1m^x2^ω(m)∏_{p | m , p ∈ ℙ}^ω(m)cos(yln(p^v_p(m))) with: z a complex number with z = x+iy and \Re(z) > 1 and x > 1 in Hardy's formula, ω(m) number of unique primes in m, v_p(m power of the prime p in m.

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