Dirichlet series and Euler products

Write $s=\sigma+it$ and interpret $n^{-s}=\exp(-s\log n)$ using the real logarithm. Its absolute value is $n^{-\sigma}$, so increasing the real part improves convergence independently of the imaginary part. The abscissa of absolute convergence is an extended real number separating an open half-plane of absolute convergence from failure to its left. Nothing universal puts that boundary at one for arbitrary coefficients. Coefficients growing like a fixed power shift the elementary comparison boundary accordingly.

Inside this open half-plane, convergence is locally uniform, and the series defines a holomorphic function. If two Dirichlet series converge absolutely at the same real part, their product can be rearranged by the product of the two integer indices. The coefficient at a given integer is exactly the Dirichlet convolution. Without absolute convergence, a convenient rearrangement may change a value or be unjustified. Analytic continuation later supplies new values of a function; it does not retroactively make the original series absolutely convergent there.

Suppose the coefficients are multiplicative and their value at one is one. Unique prime factorization expresses every coefficient as the product of its prime-power coefficients. Expanding a finite product of local series therefore sums over integers whose prime factors belong to that finite set. Absolute convergence allows the finite set of primes to expand to all primes. For completely multiplicative coefficients, each local series is geometric; ordinary multiplicativity alone does not replace an arbitrary local series by a geometric factor.

The constant-one coefficients give the Riemann zeta function, absolutely convergent for $\sigma>1$. Its local factors are reciprocal geometric denominators. On that same half-plane the product is nonzero: the logarithms of the factors form an absolutely convergent sum, since the sum of $p^{-\sigma}$ over primes is bounded by a convergent integer sum. The Möbius series is its reciprocal, and the divisor-count series is its square. These statements include their domains, rather than asserting Euler products at every continued value.

Stopping an Euler product at a prime threshold differs from stopping a Dirichlet series at an integer threshold. Even a single zeta local factor includes arbitrarily high powers of its prime. A product over two primes therefore represents infinitely many integers, while a finite integer sum contains only finitely many. Conversely, the Möbius local factors are finite, so a finite prime product becomes a finite sum over squarefree products of the selected primes. Those products can exceed the prime threshold substantially.

Logarithmic differentiation of the zeta product, justified on compact subsets of $\sigma>1$, yields $-\zeta'(s)/\zeta(s)=\sum_n\Lambda(n)n^{-s}$. Each local geometric expansion contributes every prime power with the same logarithmic weight. This is the analytic counterpart of the earlier convolution identity between the constant function, von Mangoldt weights, and the logarithm. It explains why weighted prime counts are natural for contour methods. Merely truncating the logarithmic derivative numerically supplies no theorem about the uncomputed tail.