General Dirichlet Series¶
A general Dirichlet series weights complex coefficients by exponential terms indexed by a freely chosen increasing frequency sequence.
Core Idea¶
A general Dirichlet series is a formal sum \(\sum a_n e^{-\lambda_n s}\), where the coefficients are complex and the real frequency sequence \(\lambda_n\) is nonnegative, strictly increasing, and tends to infinity. The familiar ordinary Dirichlet series uses \(\lambda_n=\log n\). With equally spaced frequencies and \(z=e^{-s}\), the expression instead resembles a power series.[^ref-b0561e034149]
Scope of Application¶
The form organizes arithmetic Dirichlet series and more general exponential series in complex analysis. With \(a_n=1\) and \(\lambda_n=\log n\), \(\sum n^{-s}\) converges absolutely for \(\Re s>1\) by the \(p\)-series comparison and diverges at real \(s=1\), giving abscissa \(1\). With \(a_n=1\), \(\lambda_n=n\) and \(q=e^{-s}\), the partial sum is \(q(1-q^N)/(1-q)\), yielding \(q/(1-q)\) for \(\Re s>0\); terms fail to vanish for \(\Re s\leq0\), so its abscissa is \(0\). These are worked specializations, not universal thresholds for all coefficients and frequencies.[ref-b0561e034149][ref-4a46b40b4272]
Clarity¶
State the frequency before transferring a result. Euler products and other arithmetic properties of ordinary \(n^{-s}\) series do not follow from arbitrary frequencies. A formal series need not converge at a finite point; its identity, convergent sum and any later analytic continuation are distinct claims.
Manages Complexity¶
Separating coefficients from the frequency shows exactly which part changes between an ordinary series and another exponential-series family. It lets the analyst test whether a theorem uses only general frequency conditions or a special arithmetic scale.
Abstract Reasoning¶
First verify the exponential term form and frequency convention. Then distinguish formal membership from ordinary, absolute or uniform convergence, locate the appropriate abscissa, and check any extra frequency assumptions before applying an analytic result.
Knowledge Transfer¶
The formal structure transfers across the logarithmic-frequency and equal-step specializations. Convergence and multiplicative conclusions transfer only with their stated hypotheses.
[^ref-b0561e034149]: Ingo Schoolmann, “On Bohr's theorem for general Dirichlet series”, arXiv:1812.04925v2 (2020), Introduction and Section 2. DOI 10.48550/arXiv.1812.04925. [^ref-4a46b40b4272]: Original recurrence-zeta research, Section 3.
Relationships to Other Abstractions¶
Current abstraction General Dirichlet Series Domain-specific
Parents (1) — more general patterns this builds on
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General Dirichlet Series presupposes Order Prime
The defining frequency sequence must be strictly increasing under real-number order.
Hierarchy paths (3) — routes to 3 parentless roots
- General Dirichlet Series → Order → Comparison → Self Checking
- General Dirichlet Series → Order → Relation
- General Dirichlet Series → Order → Set and Membership
Neighborhood in Abstraction Space¶
General Dirichlet Series sits in a sparse region of the domain-specific corpus (96th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Special Functions & Series Convergence (7 abstractions)
Nearest neighbors
- Bernoulli number — 0.78
- Formal derivative — 0.78
- Root Test — 0.78
- Abel's Limit Theorem — 0.77
- Equally Spaced Polynomial — 0.77
Computed from structural-signature embeddings · 2026-10-08