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Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof. The proof of the following lemma follow the same logic as the one of Lemma 3.3. We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof . The proof of the following lemma follow the same logic as the one of Lemma 3.3. Proof We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proofs of Auxiliary Lemmas

Table of Links Abstract and 1 Introduction 2 Background 3 On the slow growth law 4 Members of Deep Π0 1 classes 5 Strong depth is Negligible 6 Variants of Strong Depth References References [BDM23] Laurent Bienvenu, Valentino Delle Rose, and Wolfgang Merkle. Relativized depth. Theoretical Computer Science, 949:113694, 2023. [BDNM15] Laurent Bienvenu, Rodney G. Downey, Andr´e Nies, and Wolfgang Merkle. Solovay functions and their applications in algorithmic randomness. Journal of Computer and System Sciences, 81(8):1575–1591, 2015. [Ben95] Charles H. Bennett. Logical depth and physical complexity. In The Universal Turing Machine A Half-Century Survey, pages 207–235. Springer, 1995. [BHPS14] Laurent Bienvenu, Rupert H¨olzl, Christopher P. Porter, and Paul Shafer. Randomness and semi-measures. preprint, arXiv:1310.5133, 2014. [BP12] Laurent Bienvenu and Christopher P. Porter. Strong reductions in effective randomness. Theoretical Computer Science, 459:55–68, 2012. [BP16] Laurent Bienvenu and Christopher P Porter. Deep classes. Bulletin of Symbolic Logic, 22(2):249–286, 2016. [DMN17] Rod Downey, Michael McInerney, and Keng Meng Ng. Lowness and logical depth. Theoretical Computer Science, 702:23–33, 2017. [HP17] Rupert H¨olzl and Christopher P Porter. Randomness for computable measures and initial segment complexity. Annals of Pure and Applied Logic, 168(4):860–886, 2017. [JLL94] David W Juedes, James I Lathrop, and Jack H Lutz. Computational depth and reducibility. Theoretical Computer Science, 132(1-2):37–70, 1994. [Kau91] Steven M. Kautz. Degrees of random sequences. PhD thesis, Cornell University, 1991. [KHMS11] Bjørn Kjos-Hanssen, Wolfgang Merkle, and Frank Stephan. Kolmogorov complexity and the recursion theorem. Transactions of the American Mathematical Society, 363(10):5465–5480, 2011. [Lev73] Leonid A Levin. On the notion of a random sequence. In Soviet. Math. Dokl., volume 14, pages 1413–1416, 1973. [Lev84] Leonid Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15–37, 1984. [Lev13] Leonid Levin. Forbidden information. Journal of the ACM, 60(2):9, 2013. [LZ70] L. A. Levin and A. K. Zvonkin. The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms. Uspehi Mat. Nauk, 25(6(156)):85–127, 1970. [MS17] Philippe Moser and Frank Stephan. Depth, highness and dnr degrees. Discrete Mathematics & Theoretical Computer Science, 19(special issue FCT’15), 2017. [MSU98] Andrei A Muchnik, Alexei L Semenov, and Vladimir A Uspensky. Mathematical metaphysics of randomness. Theoretical Computer Science, 207(2):263–317, 1998. [Nie09] Andr´e Nies. Computability and randomness, volume 51. Oxford University Press, 2009. [SF77] Claus-Peter Schnorr and P Fuchs. General random sequences and learnable sequences. The Journal of Symbolic Logic, 42(3):329–340, 1977. Appendix A. Proof of Lemma 3 Working towards the proof of Lemma 3, we first establish the following result. We thus have the identity Then by our initial assumption on t and t′, we have The lemma follows by taking the negative logarithm of this inequality We can now prove Lemma 3. The (i)→(ii) implication is immediate. For the (ii)→(i) implication, suppose X is a sequence and h a computable increasing function such that for any time bound t. If k ∈ [h(n), h(n + 1)), we have We now apply the previous lemma with σ = X↾h(n), τ = X↾[h(n), k) and E the map such that, on input ρ, checks whether ρ has length h(n) for some n and if so returns all strings whose length is between 0 and h(n + 1) − h(n) (otherwise E(ρ) is empty). The lemma gives us a computable time bound s such that This being true for all n and all k ∈ [h(n), h(n + 1)), we have and thus X is order-deep. Finally, the (ii)↔(iii) equivalence can be established using Lemma 1(iii) and Theorem 2. This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Laurent Bienvenu;
(2) Christopher P. Porter. Photo by Matt Hardy on Unsplash Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2 Background 3 On the slow growth law 3 On the slow growth law 4 Members of Deep Π0 1 classes 4 Members of Deep Π0 1 classes 5 Strong depth is Negligible 5 Strong depth is Negligible 6 Variants of Strong Depth 6 Variants of Strong Depth References References References [BDM23] Laurent Bienvenu, Valentino Delle Rose, and Wolfgang Merkle. Relativized depth. Theoretical Computer Science, 949:113694, 2023. [BDNM15] Laurent Bienvenu, Rodney G. Downey, Andr´e Nies, and Wolfgang Merkle. Solovay functions and their applications in algorithmic randomness. Journal of Computer and System Sciences, 81(8):1575–1591, 2015. [Ben95] Charles H. Bennett. Logical depth and physical complexity. In The Universal Turing Machine A Half-Century Survey, pages 207–235. Springer, 1995. [BHPS14] Laurent Bienvenu, Rupert H¨olzl, Christopher P. Porter, and Paul Shafer. Randomness and semi-measures. preprint, arXiv:1310.5133, 2014. [BP12] Laurent Bienvenu and Christopher P. Porter. Strong reductions in effective randomness. Theoretical Computer Science, 459:55–68, 2012. [BP16] Laurent Bienvenu and Christopher P Porter. Deep classes. Bulletin of Symbolic Logic, 22(2):249–286, 2016. [DMN17] Rod Downey, Michael McInerney, and Keng Meng Ng. Lowness and logical depth. Theoretical Computer Science, 702:23–33, 2017. [HP17] Rupert H¨olzl and Christopher P Porter. Randomness for computable measures and initial segment complexity. Annals of Pure and Applied Logic, 168(4):860–886, 2017. [JLL94] David W Juedes, James I Lathrop, and Jack H Lutz. Computational depth and reducibility. Theoretical Computer Science, 132(1-2):37–70, 1994. [Kau91] Steven M. Kautz. Degrees of random sequences. PhD thesis, Cornell University, 1991. [KHMS11] Bjørn Kjos-Hanssen, Wolfgang Merkle, and Frank Stephan. Kolmogorov complexity and the recursion theorem. Transactions of the American Mathematical Society, 363(10):5465–5480, 2011. [Lev73] Leonid A Levin. On the notion of a random sequence. In Soviet. Math. Dokl., volume 14, pages 1413–1416, 1973. [Lev84] Leonid Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15–37, 1984. [Lev13] Leonid Levin. Forbidden information. Journal of the ACM, 60(2):9, 2013. [LZ70] L. A. Levin and A. K. Zvonkin. The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms. Uspehi Mat. Nauk, 25(6(156)):85–127, 1970. [MS17] Philippe Moser and Frank Stephan. Depth, highness and dnr degrees. Discrete Mathematics & Theoretical Computer Science, 19(special issue FCT’15), 2017. [MSU98] Andrei A Muchnik, Alexei L Semenov, and Vladimir A Uspensky. Mathematical metaphysics of randomness. Theoretical Computer Science, 207(2):263–317, 1998. [Nie09] Andr´e Nies. Computability and randomness, volume 51. Oxford University Press, 2009. [SF77] Claus-Peter Schnorr and P Fuchs. General random sequences and learnable sequences. The Journal of Symbolic Logic, 42(3):329–340, 1977. Appendix A. Proof of Lemma 3 Working towards the proof of Lemma 3, we first establish the following result. We thus have the identity Then by our initial assumption on t and t′, we have The lemma follows by taking the negative logarithm of this inequality We can now prove Lemma 3. The (i)→(ii) implication is immediate. For the (ii)→(i) implication, suppose X is a sequence and h a computable increasing function such that for any time bound t. If k ∈ [h(n), h(n + 1)), we have We now apply the previous lemma with σ = X↾h(n), τ = X↾[h(n), k) and E the map such that, on input ρ, checks whether ρ has length h(n) for some n and if so returns all strings whose length is between 0 and h(n + 1) − h(n) (otherwise E(ρ) is empty). The lemma gives us a computable time bound s such that This being true for all n and all k ∈ [ h ( n ), h ( n + 1)), we have n k ∈ h n h n and thus X is order-deep. X Finally, the (ii)↔(iii) equivalence can be established using Lemma 1(iii) and Theorem 2. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Laurent Bienvenu; (2) Christopher P. Porter. Authors: Authors: (1) Laurent Bienvenu; (2) Christopher P. Porter. Photo by Matt Hardy on Unsplash Matt Hardy Unsplash

Bridging Computational Notions of Depth: Working Towards the Proof of Lemma 3

Authors:
(1) Jongmin Lee, Department of Mathematical Science, Seoul National University;
(2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction 1.1 Notations and preliminaries 1.2 Prior works 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 4 Complexity lower bound 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries B Omitted proofs in Section 2 C Omitted proofs in Section 3 D Omitted proofs in Section 4 E Omitted proofs in Section 5 F Omitted proofs in Section 6 G Broader Impacts H Limitations A Preliminaries For notational unity, we use the symbol U when both V and Q can be used. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Authors: Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Notations and preliminaries 1.1 Notations and preliminaries 1.2 Prior works 1.2 Prior works 2 Anchored Value Iteration 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 3 Convergence when y=1 4 Complexity lower bound 4 Complexity lower bound 5 Approximate Anchored Value Iteration 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries A Preliminaries B Omitted proofs in Section 2 B Omitted proofs in Section 2 C Omitted proofs in Section 3 C Omitted proofs in Section 3 D Omitted proofs in Section 4 D Omitted proofs in Section 4 E Omitted proofs in Section 5 E Omitted proofs in Section 5 F Omitted proofs in Section 6 F Omitted proofs in Section 6 G Broader Impacts G Broader Impacts H Limitations H Limitations A Preliminaries For notational unity, we use the symbol U when both V and Q can be used. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Foundational Lemmas for Bellman Optimality and Anti-Optimality Operators

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 15. Evaluation of Φ1 Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as with Write where Hence First we prove that Since it follows that This yields (15.4). Let κ1(m) be given by Regarding b as an arithmetic function, for σ > 1 we have On the other hand, we can write with It follows by (15.3)-(15.5) and Proposition 14.1 that where The innermost sum above is, by the Mellin transform, equal to where This yields Hence where On substituting n = mk we can writ with Hence it follows that If (q, dl) = 1, then so that for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially It follows that the function is analytic and it satisfies for σ > 9/10. The right side of (15.14) can be rewritten as The following lemma will be proved in Appendix B. By (15.19)-(15.21) and Lemma 15.1 we obtain This yields, by (15.21), To apply (15.22) we need two lemmas which will be proved in Appendix A. Lemma 15.2. If |s − 1| < 5α, then Lemma 15.3. For σ ≥ 9/10 the function is analytic and bounded. Further we have By (4.2) and (4.3), By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain This together with Lemma 15.2 and 15.3 yields since It follows by (15.22) that By Lemma 5.8, Hence, by direct calculation, Combining these relations with (15.23) , (15.17) and (15.6) we conclude This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 15. Evaluation of Φ1 Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as with Write where Hence First we prove that Since it follows that This yields (15.4). Let κ1(m) be given by Regarding b as an arithmetic function, for σ > 1 we have b On the other hand, we can write with It follows by (15.3)-(15.5) and Proposition 14.1 that where The innermost sum above is, by the Mellin transform, equal to where This yields Hence where On substituting n = mk we can writ n = mk with Hence it follows that If ( q, dl ) = 1, then q, dl so that for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially It follows that the function is analytic and it satisfies for σ > 9/10. The right side of (15.14) can be rewritten as The following lemma will be proved in Appendix B. By (15.19)-(15.21) and Lemma 15.1 we obtain This yields, by (15.21), To apply (15.22) we need two lemmas which will be proved in Appendix A. Lemma 15.2. If |s − 1| < 5α, then Lemma 15.2. Lemma 15.3. For σ ≥ 9/10 the function Lemma 15.3. For σ ≥ 9/10 the function is analytic and bounded. Further we have is analytic and bounded. Further we have By (4.2) and (4.3), By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain This together with Lemma 15.2 and 15.3 yields since It follows by (15.22) that By Lemma 5.8, Hence, by direct calculation, Combining these relations with (15.23) , (15.17) and (15.6) we conclude This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 5. Some analytic lemmas The proofs of (5.3) and (5.4) are similar. Lemma 5.2. Let ψ and s be as in Lemma 5.1. Then Proof. The left side is By (2.6) and the Stirling formula, for |w| < 5α, Hence, for 1 ≤ j ≤ 3, The result now follows since Recall that ϑ(s) and ω(s) are given by (2.3) and (2.15) respectively. It is known that For t > 1 we have where Let and Note that Proof. By the Mellin transform (see [1], Lemma 2) we have with By the relation and Cauchy’s theorem, the proof of (5.8) is reduced to showing that for 1 ≤ j ≤ 5, where Lj denote the segments This yields (5.12) with j = 3 . As a consequence of Lemma 5.3, the Mellin transform is analytic for σ > 0. Lemma 5.4. (i). If 1/2 ≤ σ ≤ 2, then (ii). If |s − 1| < 10α, then Proof. (i). Using partial integration twice we obtain By (5.10) we have Thus some upper bounds for ∆′′(x) analogous to Lemma 5.3 can be obtained, and (i) follows. Throughout the rest of this paper we assume that (A) holds. This assumption will not be repeated in the statements of the lemmas and propositions in the sequel. The next two lemmas are weaker forms of the Deuring-Heillbronn Phenomenon. Lemma 5.7. We have Proof. The right side of the equality Lemma 5.8. If then where Proof. This follows from the relation (A) and a simple bound for L ′′(w, χ). Proof. It is known that so that By (9.1) and the condition |s − ρ| ≫ α for any ρ, so that Further, by (5.16) and Proposition 2.2 (iii), Combining theses estimates we obtain the result. In the case σ < 1/2 the proof is analogous. This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 5. Some analytic lemmas The proofs of (5.3) and (5.4) are similar. Lemma 5.2. Let ψ and s be as in Lemma 5.1. Then Lemma 5.2. Let ψ and s be as in Lemma 5.1. Then Proof . The left side is Proof By (2.6) and the Stirling formula, for |w| < 5α, Hence, for 1 ≤ j ≤ 3, The result now follows since Recall that ϑ(s) and ω(s) are given by (2.3) and (2.15) respectively. It is known that For t > 1 we have where Let and Note that Proof . By the Mellin transform (see [1], Lemma 2) we have Proof with By the relation and Cauchy’s theorem, the proof of (5.8) is reduced to showing that for 1 ≤ j ≤ 5, where Lj denote the segments This yields (5.12) with j = 3 . As a consequence of Lemma 5.3, the Mellin transform is analytic for σ > 0. Lemma 5.4 . (i). If 1/2 ≤ σ ≤ 2, then Lemma 5.4 If 1/2 ≤ σ ≤ 2, then (ii). If |s − 1| < 10α, then then Proof . (i). Using partial integration twice we obtain Proof By (5.10) we have Thus some upper bounds for ∆′′(x) analogous to Lemma 5.3 can be obtained, and (i) follows. Throughout the rest of this paper we assume that (A) holds. This assumption will not be repeated in the statements of the lemmas and propositions in the sequel. Throughout the rest of this paper we assume that (A) holds. This assumption will not be repeated in the statements of the lemmas and propositions in the sequel. The next two lemmas are weaker forms of the Deuring-Heillbronn Phenomenon. Lemma 5.7. We have Lemma 5.7. We have Proof . The right side of the equality Proof Lemma 5.8. If Lemma 5.8. If then then where where Proof . This follows from the relation Proof (A) and a simple bound for L ′′(w, χ). Proof . It is known that Proof so that By (9.1) and the condition |s − ρ| ≫ α for any ρ, so that Further, by (5.16) and Proposition 2.2 (iii), Combining theses estimates we obtain the result. In the case σ < 1/2 the proof is analogous. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Understanding the Role of Analytic Lemmas in Dirichlet L-Functions

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 4. Zeros of L(s, ψ)L(s, χψ) in Ω In this section we prove Proposition 2.2. We henceforth assume that ψ(mod p) ∈ Ψ1. This assumption will not be repeated in the statements of Lemma 4.1-4.8. We begin by proving some consequences of the inequalities (3.4)-(3.6). Lemma 4.1. Let Proof. By the Stieltjes integral we may write Hence, by partial integration, For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4). Lemma 4.2. If s ∈ Ω1, then Proof. We have Thus, similar to (4.2), by partial integration we obtain Lemma 4.3. Let We proceed to establish an approximate formula for L(s, ψ)L(s, χψ). For this purpose we first introduce a weight g(x) that will find application at various places. Let with We may write Since it follows, by changing the order of integration, that Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have Note that χψ is a primitive character (modDp). Write so that By (2.4) with θ = ψ and θ = χψ we have This yields, by Stirling’s formula, and Lemma 4.4. Let If s ∈ Ω3, then Proof. By the residue theorem, By (4.2) and (4.3), By (3.5) and partial summation, the second sum on the right side above is On the other hand, by the functional equation, for u = −σ − 1/2, and To prove (4.7) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7). To prove (4.8) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is with s ∗ = 1 + α − s¯. By partial integration, The estimate (4.9) follows by moving the contour of integration to the vertical segments and to the two connecting horizontal segments and applying (4.5) and trivial bounds for ω1(w) and the involved sum. In order to prove Proposition 2.2, it is appropriate to deal with the function By Lemma 4.2, A(s, ψ) is analytic and it has the same zeros as L(s, ψ)L(s, ψχ) in Ω1. Further, for s ∈ Ω1, we have by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that for s ∈ Ω3, where The proof of Proposition 2.2 is reduced to proving three lemmas as follows. Lemma 4.5. If then Proof. We discuss in two cases. By Lemma 4.2 and trivial estimation, Hence, by (4.5), The result now follows by (4.10). Since |B(1/2 + it, ψ)| = 1, it follows that Hence, by (4.10), Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying Proof. It suffices to show that the function A(1/2 + iγ + w, ψ) has exactly one zero inside the circle |w| = α(1 − c ′αL), counted with multiplicity. By the Rouch´e theorem, this can be reduced to proving that In either case (4.16) holds. Lemma 4.5 and 4.6 together imply the assertions (i) and (ii) of Proposition 2.2. It is also proved that the gap between any distinct zeros of A(s, ψ) in Ω is > α(1 − c ′αL). To complete the proof of the gap assertion (iii), it now suffices to prove Proof. In a way similar to the proof of Lemma 4.6, it is direct to verify that We conclude this section by giving a result which is implied in the proof of Proposition 2.2. Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Proof. It follows from Lemma 4.4 that This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 4. Zeros of L(s, ψ)L(s, χψ) in Ω In this section we prove Proposition 2.2. We henceforth assume that ψ(mod p) ∈ Ψ1. This assumption will not be repeated in the statements of Lemma 4.1-4.8. We begin by proving some consequences of the inequalities (3.4)-(3.6). Lemma 4.1. Let Lemma 4.1. Let Proof . By the Stieltjes integral we may write Proof Hence, by partial integration, For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4). Lemma 4.2. If s ∈ Ω1, then Lemma 4.2. If s ∈ Ω1, then Proof . We have Proof Thus, similar to (4.2), by partial integration we obtain Lemma 4.3. Let Lemma 4.3. Let We proceed to establish an approximate formula for L(s, ψ)L(s, χψ). For this purpose we first introduce a weight g(x) that will find application at various places. Let with We may write Since it follows, by changing the order of integration, that Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have Note that χψ is a primitive character (modDp). Write so that By (2.4) with θ = ψ and θ = χψ we have This yields, by Stirling’s formula, and Lemma 4.4. Let Lemma 4.4. Let If s ∈ Ω3, then If s ∈ Ω3, then Proof . By the residue theorem, Proof By (4.2) and (4.3), By (3.5) and partial summation, the second sum on the right side above is On the other hand, by the functional equation, for u = −σ − 1/2, and To prove (4.7) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7). To prove (4.8) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is with s ∗ = 1 + α − s¯. By partial integration, The estimate (4.9) follows by moving the contour of integration to the vertical segments and to the two connecting horizontal segments and applying (4.5) and trivial bounds for ω1(w) and the involved sum. In order to prove Proposition 2.2, it is appropriate to deal with the function By Lemma 4.2, A(s, ψ) is analytic and it has the same zeros as L(s, ψ)L(s, ψχ) in Ω1. Further, for s ∈ Ω1, we have by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that for s ∈ Ω3, where The proof of Proposition 2.2 is reduced to proving three lemmas as follows. Lemma 4.5. If Lemma 4.5. If then Proof . We discuss in two cases. Proof By Lemma 4.2 and trivial estimation, Hence, by (4.5), The result now follows by (4.10). Since |B(1/2 + it, ψ)| = 1, it follows that Hence, by (4.10), Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying Proof. It suffices to show that the function A(1/2 + iγ + w, ψ) has exactly one zero inside the circle |w| = α(1 − c ′αL), counted with multiplicity. By the Rouch´e theorem, this can be reduced to proving that In either case (4.16) holds. Lemma 4.5 and 4.6 together imply the assertions (i) and (ii) of Proposition 2.2. It is also proved that the gap between any distinct zeros of A(s, ψ) in Ω is > α(1 − c ′αL). To complete the proof of the gap assertion (iii), it now suffices to prove Proof . In a way similar to the proof of Lemma 4.6, it is direct to verify that Proof We conclude this section by giving a result which is implied in the proof of Proposition 2.2. Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Proof . It follows from Lemma 4.4 that Proof This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Detailed Lemmas on Zeros of Dirichlet L-Functions in Ω

Author:
(1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Author: Author: (1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Basic Definitions and Notations 2.1 Sets 2.1 Sets 2.2 Sequences 2.2 Sequences 2.3 Signatures and Algebras and 2.4 Categories 2.3 Signatures and Algebras and 2.4 Categories 3 Monographs and their Morphisms 3 Monographs and their Morphisms 4 Limits and Colimits 4 Limits and Colimits 5 Drawing Monographs 5 Drawing Monographs 6 Graph Structures and Typed Monographs 6 Graph Structures and Typed Monographs 7 Submonographs and Partial Morphisms 7 Submonographs and Partial Morphisms 8 Algebraic Transformations of Monographs 8 Algebraic Transformations of Monographs 9 Attributed Typed Monographs 9 Attributed Typed Monographs 10 Conclusion and References 10 Conclusion and References 5 Drawing Monographs It is sometimes necessary to name the edges in a drawing. We may then adopt the convention sometimes used for drawing diagrams in a category: the bullets are replaced by the names of the corresponding nodes, and arrows are interrupted to write their name at a place free from crossing, as in Note that no confusion is possible between the names of nodes and those of other edges, e.g., in it is clear that x and z are nodes since arrow tips point to them, and that y is the name of an edge of length 3. One particularity of monographs is that edges can be adjacent to themselves, as in Of course, knowing that a is a morphism sometimes allows to deduce the type of an edge, possibly from the types of adjacent edges. In the present case, indicating a single type would have been enough to deduce all the others. In particular applications it may be convenient to adopt completely different ways of drawing (typed) monographs. 6 Graph Structures and Typed Monographs The next lemma is central as it shows that no graph structure is omitted by the functor S if we allow sort-preserving isomorphisms of graph structures. We assume the Axiom of Choice through its equivalent formulation known as the Numeration Theorem [5]. It is therefore clear that if S were full it would be an equivalence of categories, but this is not the case as we now illustrate on graphs. The type indicated by the syntax (and consistent with the drawings of E-graphs in [2]) is of course T1. The type indicated by the syntax (and consistent with the drawings of E-graphs in [2]) is of course T1. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

A Guide to Naming and Mapping Monographs

Author:
(1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Author: Author: (1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Basic Definitions and Notations 2.1 Sets 2.1 Sets 2.2 Sequences 2.2 Sequences 2.3 Signatures and Algebras and 2.4 Categories 2.3 Signatures and Algebras and 2.4 Categories 3 Monographs and their Morphisms 3 Monographs and their Morphisms 4 Limits and Colimits 4 Limits and Colimits 5 Drawing Monographs 5 Drawing Monographs 6 Graph Structures and Typed Monographs 6 Graph Structures and Typed Monographs 7 Submonographs and Partial Morphisms 7 Submonographs and Partial Morphisms 8 Algebraic Transformations of Monographs 8 Algebraic Transformations of Monographs 9 Attributed Typed Monographs 9 Attributed Typed Monographs 10 Conclusion and References 10 Conclusion and References 4 Limits and Colimits The colimits of monographs follow the standard constructions of colimits in Sets and Graphs. Sets Graphs Theorem 4.4. The categories of Definition 3.5 are finitely co-complete. Theorem 4.4. The categories of Definition 3.5 are finitely co-complete. We next investigate the limits in categories of monographs. Products of monographs are more difficult to build than products of graphs. This is due to the fact that edges of identical length may be adjacent to edges of different lengths. We therefore see that E (A x B) is only a subset of EA x EB. Lemma 4.6. Every pair of parallel morphisms f, g : A → B has an equalizer (E, e) eq such that E is finite whenever A is finite. Lemma 4.6. Every pair of parallel morphisms f, g : A → B has an equalizer (E, e) eq such that E is finite whenever A is finite. Corollary 4.7. The monomorphisms in Monogr are the injective morphisms. Corollary 4.7. The monomorphisms in Monogr are the injective morphisms. A well-known consequence of Lemmas 4.5 and 4.6 is that all non-empty finite diagrams in Monogr have limits. Since a limit of O-monographs (resp. standard monographs) is an O-monograph (resp. standard), this holds for all categories of Definition 3.5. In particular they all have pullbacks. We shall now investigate the limits of the empty diagram in these categories, i.e., their possible terminal objects. Definition 4.8. For any set of ordinals O, let Definition 4.8. For any set of ordinals O, let Lemma 4.11. Monogr, SMonogr and FMonogr have no terminal object. Lemma 4.11. Monogr, SMonogr and FMonogr have no terminal object. Since terminal objects are limits of empty diagrams obviously these categories are not finitely complete. Theorem 4.12. O-SMonogr is finitely complete for every set of ordinals O. The categories Monogr, SMonogr and FMonogr are not finitely complete. Theorem 4.12. O SMonogr is finitely complete for every set of ordinals O. The categories Monogr SMonogr and FMonogr are not finitely complete. Proof. By Lemmas 4.5, 4.6, 4.9 and 4.11. Proof The category Graphs is also known to be adhesive, a property of pushouts and pullbacks that has important consequences on algebraic transformations (see [8]) and that we shall therefore investigate. Graphs Definition 4.13 (van Kampen squares, adhesive categories). A pushout square (A, B, C, D) is a van Kampen square if for any commutative cube Definition 4.13 A pushout square if for any commutative cube A category has pushouts along monomorphisms if all sources (A, f, g) have pushouts whenever f or g is a monomorphism. A category has pushouts along monomorphisms if all sources (A, f, g) have pushouts whenever f or g is a monomorphism. A category is adhesive if it has pullbacks, pushouts along monomorphisms and all such pushouts are van Kampen squares. A category is adhesive if it has pullbacks, pushouts along monomorphisms and all such pushouts are van Kampen squares. As in the proof that Graphs is adhesive, we will use the fact that the category Sets is adhesive. Graphs Sets Lemma 4.14. E reflects isomorphisms. Lemma 4.14. reflects isomorphisms. A side consequence is that Monogr is balanced, i.e., if f is both a monomorphism and an epimorphism, then by Corollaries 4.3 and 4.7 f is bijective, hence is an isomorphism. More important is that we can use [7, Theorem 24.7], i.e., that a faithful and isomorphism reflecting functor from a category that has some limits or colimits and preserves them, also reflects them. Monogr Lemma 4.15. E preserves and reflects finite colimits. Lemma 4.15. preserves and reflects finite colimits. Proof. It is easy to see from the proofs of Lemmas 4.1 and 4.2 that E preserves both coproducts and coequalizers, so that E preserves all finite co-limits and hence also reflects them. Proof This is particularly true for pushouts. The situation for pullbacks is more complicated since E does not preserve products. Lemma 4.16. E preserves and reflects pullbacks. preserves and reflects pullbacks. Theorem 4.17. The categories of Definition 3.5 are adhesive. Theorem 4.17. The categories of Definition 3.5 are adhesive. Proof. The existence of pullbacks and pushouts is already established. In any of these categories a commutative cube built on a pushout along a monomorphism as bottom face and with pullbacks as back faces, has an underlying cube in Sets that has the same properties by Corollary 4.7, Lemmas 4.15 and 4.16. Since Sets is an adhesive category (see [8]) the underlying bottom face is a van Kampen square, hence such is the bottom face of the initial cube by Lemmas 4.15 and 4.16. Proof Sets Sets This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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