论文标题
Dirichlet卷积的重构
Refactorisation of the Dirichlet convolution
论文作者
论文摘要
我们提出了一种考虑Dirichlet卷积的新方法，以使我们构建一个由分解中涉及的操作产生的环。我们将以这项工作中发现的某些身份总结。结果的应用使我们对以下硬度公式进行了概括：$$ζ（x）^{2} =ζ（2x）\ sum_ {m = 1}^{+\ frac} \ frac {2^{ω（ω（m）}}}}}}} {m^{x}} {x}} $ | ζ（2x）\ sum_ {m = 1}^{+\ infty} \ frac {1} {m^{x}}} 2^{ω（m）} \ prod_ {p | m，p \ in \ mathbb {p}}}^{ω（m）} \ cos（y \ ln（y \ ln（p^{v_ {p}（m）}））$$ with：$ z $ a $ z = z = x+iy $和$ yy $ and $（z）> 1 $和x $ y（z）> 1 $ and $ pr的复杂数字， $ v_ {p}（$ m $ prime $ p $的m $ power。
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: $$ζ(x)^{2} = ζ(2x)\sum_{m=1}^{+\infty} \frac{2^{ω(m)}}{m^{x}}$$ which is: $$|ζ(z)|^{2} = ζ(2x)\sum_{m=1}^{+\infty}\frac{1}{m^{x}}2^{ω(m)}\prod_{p | m , p \in \mathbb{P}}^{ω(m)}\cos(y\ln(p^{v_{p}(m)}))$$ 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 $m$.
