10 Search Results for "Eberl, Manuel"

Document
DOI: 10.4230/LIPIcs.ITP.2025.2
A Formal Analysis of Algorithms for Matroids and Greedoids

Authors: Mohammad Abdulaziz, Thomas Ammer, Shriya Meenakshisundaram, and Adem Rimpapa

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
We present a formal analysis, in Isabelle/HOL, of optimisation algorithms for matroids, which are useful generalisations of combinatorial structures that occur in optimisation, and greedoids, which are a generalisation of matroids. Although some formalisation work has been done earlier on matroids, our work here presents the first formalisation of results on greedoids, and many results we formalise in relation to matroids are also formalised for the first time in this work. We formalise the analysis of a number of optimisation algorithms for matroids and greedoids. We also derive from those algorithms executable implementations of Kruskal’s algorithm for computing optimal spanning trees, an algorithm for maximum cardinality matching for bi-partite graphs, and Prim’s algorithm for computing minimum weight spanning trees.
Cite as

Mohammad Abdulaziz, Thomas Ammer, Shriya Meenakshisundaram, and Adem Rimpapa. A Formal Analysis of Algorithms for Matroids and Greedoids. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{abdulaziz_et_al:LIPIcs.ITP.2025.2,
  author =	{Abdulaziz, Mohammad and Ammer, Thomas and Meenakshisundaram, Shriya and Rimpapa, Adem},
  title =	{{A Formal Analysis of Algorithms for Matroids and Greedoids}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{2:1--2:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.2},
  URN =		{urn:nbn:de:0030-drops-246012},
  doi =		{10.4230/LIPIcs.ITP.2025.2},
  annote =	{Keywords: Matroids, Greedoids, Combinatorial Optimisation, Graph Algorithms, Isabelle/HOL, Formal Verification}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.35
Verifying an Efficient Algorithm for Computing Bernoulli Numbers

Authors: Manuel Eberl and Peter Lammich

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
The Bernoulli numbers B_k are a sequence of rational numbers that is ubiquitous in mathematics, but difficult to compute efficiently (compared to e.g. approximating π). In 2008, Harvey gave the currently fastest known practical way for computing them: his algorithm computes B_k mod p in time O(p log^{1 + o(1)} p). By doing this for O(k) many small primes p in parallel and then combining the results with the Chinese Remainder Theorem, one recovers the value of B_k as a rational number in O(k² log^{2 + o(1)} k) time. One advantage of this approach is that the expensive part of the algorithm is highly parallelisable and has very low memory requirements. This algorithm still holds the world record with its computation of B_{10⁸}. We give a verified efficient LLVM implementation of this algorithm. This was achieved by formalising the necessary mathematical background theory in Isabelle/HOL, proving an abstract version of the algorithm correct, and refining this abstract version down to LLVM using Lammich’s Isabelle-LLVM framework, including many low-level optimisations. The performance of the resulting LLVM code is comparable with Harvey’s original unverified and hand-optimised C++ implementation.
Cite as

Manuel Eberl and Peter Lammich. Verifying an Efficient Algorithm for Computing Bernoulli Numbers. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eberl_et_al:LIPIcs.ITP.2025.35,
  author =	{Eberl, Manuel and Lammich, Peter},
  title =	{{Verifying an Efficient Algorithm for Computing Bernoulli Numbers}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{35:1--35:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.35},
  URN =		{urn:nbn:de:0030-drops-246331},
  doi =		{10.4230/LIPIcs.ITP.2025.35},
  annote =	{Keywords: Bernoulli numbers, LLVM, verification, Isabelle, Chinese remainder theorem, modular arithmetic, Montgomery arithmetic}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.18
Formalising New Mathematics in Isabelle: Diagonal Ramsey

Authors: Lawrence C. Paulson

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
The formalisation of mathematics is becoming routine, but its value to research mathematicians remains unproven. There are few examples of using proof assistants to verify new work. This paper reports the formalisation - inspired by a Lean one by Bhavik Mehta - of a major new result [Marcelo Campos et al., 2023] about Ramsey numbers. One unexpected finding was a heavy role for computer algebra techniques.
Cite as

Lawrence C. Paulson. Formalising New Mathematics in Isabelle: Diagonal Ramsey. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{paulson:LIPIcs.ITP.2025.18,
  author =	{Paulson, Lawrence C.},
  title =	{{Formalising New Mathematics in Isabelle: Diagonal Ramsey}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.18},
  URN =		{urn:nbn:de:0030-drops-246163},
  doi =		{10.4230/LIPIcs.ITP.2025.18},
  annote =	{Keywords: Isabelle, formalisation of mathematics, Ramsey’s theorem, computer algebra}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.ITP.2024.40
Formalising Half of a Graduate Textbook on Number Theory (Short Paper)

Authors: Manuel Eberl, Anthony Bordg, Lawrence C. Paulson, and Wenda Li

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
Apostol’s Modular Functions and Dirichlet Series in Number Theory [Tom M. Apostol, 1990] is a graduate text covering topics such as elliptic functions, modular functions, approximation theorems and general Dirichlet series. It relies on complex analysis, winding numbers, the Riemann ζ function and Laurent series. We have formalised several chapters and can comment on the sort of gaps found in pedagogical mathematics. Proofs are available from https://github.com/Wenda302/Number_Theory_ITP2024.
Cite as

Manuel Eberl, Anthony Bordg, Lawrence C. Paulson, and Wenda Li. Formalising Half of a Graduate Textbook on Number Theory (Short Paper). In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 40:1-40:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eberl_et_al:LIPIcs.ITP.2024.40,
  author =	{Eberl, Manuel and Bordg, Anthony and Paulson, Lawrence C. and Li, Wenda},
  title =	{{Formalising Half of a Graduate Textbook on Number Theory}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{40:1--40:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.40},
  URN =		{urn:nbn:de:0030-drops-207686},
  doi =		{10.4230/LIPIcs.ITP.2024.40},
  annote =	{Keywords: Isabelle/HOL, number theory, complex analysis, formalisation of mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.4
Use and Abuse of Instance Parameters in the Lean Mathematical Library

Authors: Anne Baanen

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean’s mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.
Cite as

Anne Baanen. Use and Abuse of Instance Parameters in the Lean Mathematical Library. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baanen:LIPIcs.ITP.2022.4,
  author =	{Baanen, Anne},
  title =	{{Use and Abuse of Instance Parameters in the Lean Mathematical Library}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.4},
  URN =		{urn:nbn:de:0030-drops-167131},
  doi =		{10.4230/LIPIcs.ITP.2022.4},
  annote =	{Keywords: formalization of mathematics, dependent type theory, typeclasses, algebraic hierarchy, Lean prover}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.21
Formalization of Randomized Approximation Algorithms for Frequency Moments

Authors: Emin Karayel

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
In 1999 Alon et al. introduced the still active research topic of approximating the frequency moments of a data stream using randomized algorithms with minimal space usage. This includes the problem of estimating the cardinality of the stream elements - the zeroth frequency moment. Higher-order frequency moments provide information about the skew of the data stream which is, for example, critical information for parallel processing. (The k-th frequency moment of a data stream is the sum of the k-th powers of the occurrence counts of each element in the stream.) They introduce both lower bounds and upper bounds on the space complexity of the problems, which were later improved by newer publications. The algorithms have guaranteed success probabilities and accuracies without making any assumptions on the input distribution. They are an interesting use case for formal verification because their correctness proofs require a large body of deep results from algebra, analysis and probability theory. This work reports on the formal verification of three algorithms for the approximation of F₀, F₂ and F_k for k ≥ 3. The results include the identification of significantly simpler algorithms with the same runtime and space complexities as the previously known ones as well as the development of several reusable components, such as a formalization of universal hash families, amplification methods for randomized algorithms, a model for one-pass data stream algorithms or a generic flexible encoding library for the verification of space complexities.
Cite as

Emin Karayel. Formalization of Randomized Approximation Algorithms for Frequency Moments. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{karayel:LIPIcs.ITP.2022.21,
  author =	{Karayel, Emin},
  title =	{{Formalization of Randomized Approximation Algorithms for Frequency Moments}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.21},
  URN =		{urn:nbn:de:0030-drops-167308},
  doi =		{10.4230/LIPIcs.ITP.2022.21},
  annote =	{Keywords: Formal Verification, Isabelle/HOL, Randomized Algorithms, Frequency Moments}
}
Document
DOI: 10.4230/LIPIcs.ITP.2021.22
Formalization of Basic Combinatorics on Words

Authors: Štěpán Holub and Štěpán Starosta

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
Combinatorics on Words is a rather young domain encompassing the study of words and formal languages. An archetypal example of a task in Combinatorics on Words is to solve the equation x ⋅ y = y ⋅ x, i.e., to describe words that commute. This contribution contains formalization of three important classical results in Isabelle/HOL. Namely i) the Periodicity Lemma (a.k.a. the theorem of Fine and Wilf), including a construction of a word proving its optimality; ii) the solution of the equation x^a ⋅ y^b = z^c with 2 ≤ a,b,c, known as the Lyndon-Schützenberger Equation; and iii) the Graph Lemma, which yields a generic upper bound on the rank of a solution of a system of equations. The formalization of those results is based on an evolving toolkit of several hundred auxiliary results which provide for smooth reasoning within more complex tasks.
Cite as

Štěpán Holub and Štěpán Starosta. Formalization of Basic Combinatorics on Words. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{holub_et_al:LIPIcs.ITP.2021.22,
  author =	{Holub, \v{S}t\v{e}p\'{a}n and Starosta, \v{S}t\v{e}p\'{a}n},
  title =	{{Formalization of Basic Combinatorics on Words}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.22},
  URN =		{urn:nbn:de:0030-drops-139177},
  doi =		{10.4230/LIPIcs.ITP.2021.22},
  annote =	{Keywords: combinatorics on words, formalization, Isabelle/HOL}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.11
A Verified and Compositional Translation of LTL to Deterministic Rabin Automata

Authors: Julian Brunner, Benedikt Seidl, and Salomon Sickert

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
We present a formalisation of the unified translation approach from linear temporal logic (LTL) to omega-automata from [Javier Esparza et al., 2018]. This approach decomposes LTL formulas into "simple" languages and allows a clear separation of concerns: first, we formalise the purely logical result yielding this decomposition; second, we develop a generic, executable, and expressive automata library providing necessary operations on automata to re-combine the "simple" languages; third, we instantiate this generic theory to obtain a construction for deterministic Rabin automata (DRA). We extract from this particular instantiation an executable tool translating LTL to DRAs. To the best of our knowledge this is the first verified translation of LTL to DRAs that is proven to be double-exponential in the worst case which asymptotically matches the known lower bound.
Cite as

Julian Brunner, Benedikt Seidl, and Salomon Sickert. A Verified and Compositional Translation of LTL to Deterministic Rabin Automata. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brunner_et_al:LIPIcs.ITP.2019.11,
  author =	{Brunner, Julian and Seidl, Benedikt and Sickert, Salomon},
  title =	{{A Verified and Compositional Translation of LTL to Deterministic Rabin Automata}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.11},
  URN =		{urn:nbn:de:0030-drops-110664},
  doi =		{10.4230/LIPIcs.ITP.2019.11},
  annote =	{Keywords: Automata Theory, Automata over Infinite Words, Deterministic Automata, Linear Temporal Logic, Model Checking, Verified Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.15
Formalizing the Solution to the Cap Set Problem

Authors: Sander R. Dahmen, Johannes Hölzl, and Robert Y. Lewis

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of F^n_q with no three-term arithmetic progression. This problem has received much mathematical attention, particularly in the case q = 3, where it is commonly known as the cap set problem. Ellenberg and Gijswijt’s proof was published in the Annals of Mathematics and is noteworthy for its clever use of elementary methods. This paper describes a formalization of this proof in the Lean proof assistant, including both the general result in F^n_q and concrete values for the case q = 3. We faithfully follow the pen and paper argument to construct the bound. Our work shows that (some) modern mathematics is within the range of proof assistants.
Cite as

Sander R. Dahmen, Johannes Hölzl, and Robert Y. Lewis. Formalizing the Solution to the Cap Set Problem. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dahmen_et_al:LIPIcs.ITP.2019.15,
  author =	{Dahmen, Sander R. and H\"{o}lzl, Johannes and Lewis, Robert Y.},
  title =	{{Formalizing the Solution to the Cap Set Problem}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.15},
  URN =		{urn:nbn:de:0030-drops-110703},
  doi =		{10.4230/LIPIcs.ITP.2019.15},
  annote =	{Keywords: formal proof, combinatorics, cap set problem, Lean}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.16
Nine Chapters of Analytic Number Theory in Isabelle/HOL

Authors: Manuel Eberl

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
In this paper, I present a formalisation of a large portion of Apostol’s Introduction to Analytic Number Theory in Isabelle/HOL. Of the 14 chapters in the book, the content of 9 has been mostly formalised, while the content of 3 others was already mostly available in Isabelle before. The most interesting results that were formalised are: - The Riemann and Hurwitz zeta functions and the Dirichlet L functions - Dirichlet’s theorem on primes in arithmetic progressions - An analytic proof of the Prime Number Theorem - The asymptotics of arithmetical functions such as the prime omega function, the divisor count sigma_0(n), and Euler’s totient function phi(n)
Cite as

Manuel Eberl. Nine Chapters of Analytic Number Theory in Isabelle/HOL. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{eberl:LIPIcs.ITP.2019.16,
  author =	{Eberl, Manuel},
  title =	{{Nine Chapters of Analytic Number Theory in Isabelle/HOL}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.16},
  URN =		{urn:nbn:de:0030-drops-110714},
  doi =		{10.4230/LIPIcs.ITP.2019.16},
  annote =	{Keywords: Isabelle, theorem proving, analytic number theory, number theory, arithmetical function, Dirichlet series, prime number theorem, Dirichlet’s theorem, zeta function, L functions}
}
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