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Journal articles on the topic 'Trivial zeros'

Author: Grafiati
Published: 30 November 2024
Last updated: 19 July 2025

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1

Bayer, Pilar. "La hipòtesi de Riemann: El gran repte pendent." Mètode Revista de difusió de la investigació, no. 8 (June 5, 2018): 35. http://dx.doi.org/10.7203/metode.0.8903.

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The Riemann hypothesis is an unproven statement referring to the zeros of the Riemann zeta function. Bernhard Riemann calculated the first six non-trivial zeros of the function and observed that they were all on the same straight line. In a report published in 1859, Riemann stated that this might very well be a general fact. The Riemann hypothesis claims that all non-trivial zeros of the zeta function are on the the line x = 1/2. The more than ten billion zeroes calculated to date, all of them lying on the critical line, coincide with Riemann’s suspicion, but no one has yet been able to prove that the zeta function does not have non-trivial zeroes outside of this line.
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2

Lam Kai, Shun. "A Verification of Riemann Non-Trivial Zeros by Complex Analysis by Matlab™ Computation." European Journal of Statistics and Probability 11, no. 1 (2023): 69–83. http://dx.doi.org/10.37745/ejsp.2013/vol11n16983.

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With my most recent paper, I tried to prove the Riemann Hypothesis by catching out those contradictory parts of the non-trivial zeros. In the present paper, I will try to verify these known values of Riemann nontrivial zeros by first using U.S.A. Matlab coding with a list of well-organized complex analysis theories. At the same time, as the major core of my verification is just a mono-direction one (i.e. there may be a possibility of the missing non-trivial zeros although the residue value is zero), hence this author try to solve such problem by assuming that there are some other zeros existing between the two known zeros but the contradiction arises – as singularity implies the residue has a value with a multiple of 2πi. In addition, this author also apply the ingenious design (or a hybrid skill) with Feynman technique and Integration by parts to solve a special zeta function integral. Next, this author finds that one may consider those non-trivial zeros as a Fourier transform (or an impulse) between other normal complex numbers. The result is consistent with my previous papers in quantum physics [23], [25] for the electron jumps or reverse. Hence, we may get the (dirac) delta equation for Riemann Zeta. Then we may formulate our quantum circuit & computer. Finally, this author concludes all findings with an algorithm for searching, finer and checking the non-trivial zeros like below: Step 1: Use the computer software with some suitable program codes for an elementary search of feasible non-trivial zeta values among the closed real-complex plane interval – Method Matlab Simulation for searching zeta zeros; Step 2: Substitute back the values laying in the contour interval for zeta as found in Step 1 into the limit of ln(zeta(z))/((zeta'(z)) ) in order to adjust the answer in a finer and accurate way (just like the case of Newton’s method etc) with more decimal digitals – Method Ingenious Design for finer the zeta zero’s values; Step 3: Employ the Cauchy Residue Theorem for a check and hence confirm the previous found non-trivial zeta roots’ uniqueness without any zeta zeros laying in between the two consecutive zeta roots – Method Cauchy’s Residue for checking those already found zeta zeros.
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3

Pushkarev, Petr. "Constant quality of the Riemann zeta's non-trivial zeros." Global Journal of Pure and Applied Mathematics 13, no. 6 (2017): 1987–92. https://doi.org/10.5281/zenodo.822059.

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In this article we are closely examining Riemann zeta function's non-trivial zeros. Especially, we examine real part of non-trivial zeros. Real part of Riemann zeta function's non-trivial zeros is considered in the light of constant quality of such zeros. We propose and prove a theorem of this quality. We also uncover a definition phenomenons of zeta and Riemann xi functions. In conclusion and as an conclusion we observe Riemann hypothesis in perspective of our researches.
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4

HASSEN, ABDUL, and HIEU D. NGUYEN. "A ZERO-FREE REGION FOR HYPERGEOMETRIC ZETA FUNCTIONS." International Journal of Number Theory 07, no. 04 (2011): 1033–43. http://dx.doi.org/10.1142/s1793042111004678.

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This paper investigates the location of "trivial" zeros of some hypergeometric zeta functions. Analogous to Riemann's zeta function, we demonstrate that they possess a zero-free region on a left-half complex plane, except for infinitely many zeros regularly spaced on the negative real axis.
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5

Platt, David J. "Isolating some non-trivial zeros of zeta." Mathematics of Computation 86, no. 307 (2017): 2449–67. http://dx.doi.org/10.1090/mcom/3198.

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6

Silva, Sergio Da. "The Riemann Hypothesis: A Fresh and Experimental Exploration." Journal of Advances in Mathematics and Computer Science 39, no. 4 (2024): 100–112. http://dx.doi.org/10.9734/jamcs/2024/v39i41885.

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This research proposes a new approach to the Riemann Hypothesis, focusing on the interplay between prime gaps and the non-trivial zeros of the Riemann Zeta function. Utilizing various statistical models and experimental analysis techniques, three important insights are uncovered: 1) Granger causality tests reveal a predictive relationship in which past non-trivial zeros may predict future prime gaps; 2) Complex, nonlinear interactions between prime gaps and non-trivial zeros are identified, challenging simple linear correlations; and 3) Causal network analysis reveals intricate feedback-loop relationships. These findings contribute to a better understanding of prime number distribution and the Zeta function, opening up novel possibilities for further mathematical research. The study aims to motivate mathematicians towards a proof or disproof of the Riemann Hypothesis.
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7

Ghisa, Dorin. "On the Non-Trivial Zeros of Dirichlet Functions." Advances in Pure Mathematics 11, no. 03 (2021): 187–204. http://dx.doi.org/10.4236/apm.2021.113014.

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8

Suman, Shekhar, and Raman Kumar Das. "A NOTE ON AN EQUIVALENT OF THE RIEMANN HYPOTHESIS." JOURNAL OF RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES 10, no. 01 (2022): 97–102. http://dx.doi.org/10.56827/jrsmms.2022.1001.8.

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In this manuscript we denote by P ρ a sum over the non trivial zeros of Riemann zeta function (or over the zeros of Riemann’s xi function), where the zeros of multiplicity k are counted k times. We prove a result that the Riemann Hypothesis is true if and only if
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9

Apostol, Tom M. "Note on the Trivial Zeros of Dirichlet L-Functions." Proceedings of the American Mathematical Society 94, no. 1 (1985): 29. http://dx.doi.org/10.2307/2044944.

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10

Rohrlich, David E. "Scarcity and abundance of trivial zeros in division towers." Journal of Algebraic Geometry 17, no. 4 (2008): 643–75. http://dx.doi.org/10.1090/s1056-3911-08-00462-1.

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11

Benois, Denis. "Trivial zeros of -adic -functions at near-central points." Journal of the Institute of Mathematics of Jussieu 13, no. 3 (2013): 561–98. http://dx.doi.org/10.1017/s1474748013000261.

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AbstractUsing the$\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of$p$-adic$L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.
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12

Apostol, Tom M. "Note on the trivial zeros of Dirichlet $L$-functions." Proceedings of the American Mathematical Society 94, no. 1 (1985): 29. http://dx.doi.org/10.1090/s0002-9939-1985-0781049-8.

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13

Sekatskii, Sergey. "On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros." Symmetry 16, no. 3 (2024): 326. http://dx.doi.org/10.3390/sym16030326.

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Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.
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14

soltan, Shaimaa said. "New Odd Numbers Identity and the None-trivial Zeros of Zeta Function." Journal of Mathematics Research 15, no. 2 (2023): 74. http://dx.doi.org/10.5539/jmr.v15n2p74.

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This paper is going to introduce a new identity unit circle function for complex plane specific for odd numbers.
 
 Second, we are going to show some properties of these new unit Identity function.
 
 Third, use this new unit Identity function to study the distribution of odd roots for sin term in zeta function but using the new identity function not Euler Identity to explain Riemann conjunction about the critical strip line and the none-trivial zeros along Re(S) = 0.5.
 
 Also, In an Introductory Analysis for the geometric functions Sin and Cos, we will visualize the inverse of geometric function Sin.
 
 Riemann's functional equation
 
 
 
 
 
 Then Zeta function will be zero
 
 
 At             is Zero for any complex number S.
 If exponential term is zero also when S = S + 0.5 where S is any complex number.
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15

BERGWEILER, WALTER, ALEXANDRE EREMENKO, and AIMO HINKKANEN. "Entire functions with two radially distributed values." Mathematical Proceedings of the Cambridge Philosophical Society 165, no. 1 (2017): 93–108. http://dx.doi.org/10.1017/s0305004117000305.

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AbstractWe study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.
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16

Thalassinakis, Emmanuel. "An In-Depth Investigation of the Riemann Zeta Function Using Infinite Numbers." Mathematics 13, no. 9 (2025): 1483. https://doi.org/10.3390/math13091483.

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This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved.
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17

Wong, Bertrand. "Non-trivial zeros of riemann zeta function and riemann hypothesis." Bulletin of Pure & Applied Sciences- Mathematics and Statistics 41e, no. 1 (2022): 88–99. http://dx.doi.org/10.5958/2320-3226.2022.00013.3.

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18

Hemingway, Xian. "The Generalized Riemann Hypothesis on elliptic complex fields." AIMS Mathematics 8, no. 11 (2023): 25772–803. http://dx.doi.org/10.3934/math.20231315.

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<abstract><p>In this paper, we will introduce a new algebraic system called the elliptic complex, and consider the distribution of zeros of the function $ L(s, \chi) $ in the corresponding complex plane. The key to this article is to discover the limiting case of the Generalized Riemann Hypothesis on elliptic complex fields and, taking a series of elliptic complex fields as variables, to study the ordinary properties of their distributions about the non-trivial zeros of $ L(s, \chi) $. It is on the basis of these considerations that we will draw the following conclusions. First, the zeros of the function $ L(s, \chi) $ on any two elliptic complex planes correspond one-to-one. Then, all non-trivial zeros of the $ L(s, \chi) $ function on each elliptic complex plane are distributed on the critical line $ \Re(s) = {1\over 2} $ due to the critical case of the Generalized Riemann Hypothesis. Ultimately we proved the Generalized Riemann Hypothesis.</p></abstract>
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19

FARAG, HANY M. "DIRICHLET TRUNCATIONS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP POSSESS ZEROS NEAR EVERY VERTICAL LINE." International Journal of Number Theory 04, no. 04 (2008): 653–62. http://dx.doi.org/10.1142/s1793042108001596.

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We study the zeros of the finite truncations of the alternating Dirichlet series expansion of the Riemann zeta function in the critical strip. We do this with an (admittedly highly) ambitious goal in mind. Namely, that this series converges to the zeta function (up to a trivial term) in the critical strip and our hope is that if we can obtain good estimates for the zeros of these approximations it may be possible to generalize some of the results to zeta itself. This paper is a first step towards this goal. Our results show that these finite approximations have zeros near every vertical line (so no vertical strip in this region is zero-free). Furthermore, we give upper bounds for the imaginary parts of the zeros (the real parts are pinned). The bounds are numerically very large. Our tools are: the inverse mapping theorem (for a perturbative argument), the prime number theorem (for counting primes), elementary geometry (for constructing zeros of a related series), and a quantitative version of Kronecker's theorem to go from one series to another.
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20

TOURIGNY, DAVID S. "RIEMANN ZEROS AND THE INVERSE PHASE PROBLEM." Modern Physics Letters B 27, no. 26 (2013): 1350187. http://dx.doi.org/10.1142/s021798491350187x.

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Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centro-symmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.
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21

Shavala, O. "ON THE CONSTRUCTION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS ACCORDING TO GIVEN SEQUENCES OF ZEROS AND CRITICAL POINTS." Bukovinian Mathematical Journal 11, no. 1 (2023): 134–37. http://dx.doi.org/10.31861/bmj2023.01.12.

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A part of the theory of differential equations in the complex plane $\mathbb C$ is the study of their solutions. To obtain them sometimes researchers can use local expand of solution in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton - Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable. One of the directions of the theory of differential equations in the complex plane $\mathbb C$ is the construction a function $f$ according given sequence of zeros or poles, zeros of the derivative $f'$ and then find a differential equation for which this function be solution. Some authors studied sequences of zeros of solutions of the linear differential equation \begin{equation*} f''+Af=0, \end{equation*} where $A$ is entire or analytic function in a disk ${\rm \{ z:|z| < 1\} }$. In addition to the case when the above-mentioned differential equation has the non-trivial solution with given zero-sequences it is possible for consideration the case, when this equation has a solution with a given sequence of zeros (poles) and critical points. In this article we consider the question when the above-mentioned differential equation has the non-trivial solution $f$ such that $f^{1/\alpha}$, $\alpha \in {\mathbb R}\backslash \{ 0;-1\} $ is meromorphic function without zeros with poles in given sequence and the derivative of solution $f'$ has zeros in other given sequence, where $A$ is meromorphic function. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions. Further we consider the question about construction of entire solution $f$ of the differential equation \begin{equation*} f^{(n)} +Af^{m} =0, \quad n,m\in {\mathbb N}, \end{equation*} where $A$ is meromorphic function such that $f$ has zeros in given sequence and the derivative of solution $f'$ has zeros in other given sequence.
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22

Özlük, Ali E., and C. Snyder. "Small zeros of quadratic L-functions." Bulletin of the Australian Mathematical Society 47, no. 2 (1993): 307–19. http://dx.doi.org/10.1017/s0004972700012545.

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We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour ofas D → ∞. Here denotes the sum over the non-trivial zeros p = 1/2 + iγ of the Dirichlet L-function L(s, χd), and χd = () is the Kronecker symbol. The outer sum is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for
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23

Deza, Elena, and Lidiya Varukhina. "REPRESENTATIONS OF ARITHMETIC SUMS OVER NON-TRIVIAL ZEROS OF THE ZETA FUNCTION." Asian-European Journal of Mathematics 01, no. 04 (2008): 509–19. http://dx.doi.org/10.1142/s1793557108000412.

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We obtain representations over non-trivial zeros of zeta function for two arithmetic functions ψ1(x) = ∑n≤x (x - n)Λ(n) and [Formula: see text]. This result is similar to classical representations of such kind for the Chebyshev functionψ(x) = ∑n≤x Λ(n).
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24

Lindner, A. "Non-trivial zeros of the Wigner (3j) and Racah (6j) coefficients." Journal of Physics A: Mathematical and General 18, no. 15 (1985): 3071–72. http://dx.doi.org/10.1088/0305-4470/18/15/029.

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25

KANEKO, SATORU, HIDEYUKI SAWANAKA, TAKAYA SHINGAI, MORIMITSU TANIMOTO, and KOICHI YOSHIOKA. "FLAVOR SYMMETRY AND VACUUM ALIGNED MASS TEXTURES." International Journal of Modern Physics E 16, no. 05 (2007): 1427–36. http://dx.doi.org/10.1142/s0218301307006782.

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A texture-zeros is an approach to reduce the number of free parameters in Yukawa couplings and it is one of the most attractive ones. In our paper, we discuss the origin of zero-structure in texture-zeros by S3 flavor symmetry approach. Some of electroweak doublet Higgs fields have vanishing vacuum expectation value (VEV) which leads to vanishing elements in quark and lepton mass matrices. Then, the structure of supersymmetric scalar potential is analyzed and Higgs fields have non-trivial S3 charges. As a prediction of our paper, a lower bound of a MNS matrix element, Ue3 ≥ 0.04, is obtained. The suppression of flavor-changing neutral currents (FCNC) mediated by the Higgs fields is discussed and lower bounds of the Higgs masses are derived.
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26

Ting, John Y. C. "Key Role of Dimensional Analysis Homogeneity in Proving Riemann Hypothesis and Providing Explanations on the Closely Related Gram Points." Journal of Mathematics Research 8, no. 4 (2016): 1. http://dx.doi.org/10.5539/jmr.v8n4p1.

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Riemann zeta function is the famous complex number infinite series consisting of a real and an imaginary part. Non-trivial zeros and Gram points are best seen as mathematically derived entities of this function when its variable Sigma has a value of $\frac{1}{2}$. The presence [but not the actual locations] of the complete set of infinite non-trivial zeros is characterized by the criterion that the sum total of the simultaneous real and imaginary parts in Riemann zeta function equates to zero. In an identical manner this slightly altered criterion for the presence [but not the actual locations] of the complete set of infinite Gram points is that this 'sum total' now refer to the lesser requirement that only the individual imaginary part in Riemann zeta function equates to zero. The key role played by Dimensional analysis homogeneity to rigorously prove Riemann conjecture/hypothesis has been fully outlined in our landmark research paper published earlier on Page 9 - 21 in the preceding Volume 8, Number 3, June 2016 issue of this journal. Those resulting methodology previously employed by us are now mathematically used in an analogical procedure to delineate its role in successfully supplying crucial explanations for Gram points. In this research article, we use the notation \{Non-critical lines\}-Gram points to signify those 'near-identical' (virtual) Gram points when Sigma value is not $\frac{1}{2}$.
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27

Wang, Wei, Yanjie Zhu, Zhuoxu Cui, and Dong Liang. "Is Each Layer Non-trivial in CNN? (Student Abstract)." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 18 (2021): 15915–16. http://dx.doi.org/10.1609/aaai.v35i18.17954.

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Convolutional neural network (CNN) models have achieved great success in many fields. With the advent of ResNet, networks used in practice are getting deeper and wider. However, is each layer non-trivial in networks? To answer this question, we trained a network on the training set, then we replace the network convolution kernels with zeros and test the result models on the test set. We compared experimental results with baseline and showed that we can reach similar or even the same performances. Although convolution kernels are the cores of networks, we demonstrate that some of them are trivial and regular in ResNet.
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28

Kiguradze, Ivan. "Beurling–Borg Type Theorem for Two-Dimensional Linear Differential Systems." gmj 15, no. 4 (2008): 677–82. http://dx.doi.org/10.1515/gmj.2008.677.

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Abstract For the two-dimensional linear differential system with Lebesgue integrable coefficients 𝑝𝑖𝑘 : [𝑎, 𝑏] → ℝ (𝑖 = 1, 2), a Beurling–Borg type theorem is proved on an upper estimate of the number of zeros of an arbitrary non-trivial solution.
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29

Csoka, Levente. "The Fourier Transform of the Non-Trivial Zeros of the Zeta Function." Journal of Advances in Applied & Computational Mathematics 4, no. 1 (2018): 23–25. http://dx.doi.org/10.15377/2409-5761.2017.04.4.

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30

Somvanshi, Manvendra. "On Trivial Zeros of Zeta Function and Problems with the Analytic Continuation." International Journal of Mathematics Trends and Technology 55, no. 6 (2018): 430–33. http://dx.doi.org/10.14445/22315373/ijmtt-v55p557.

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31

Laurinčikas, Antanas. "Non-trivial zeros of the Riemann zeta-function and joint universality theorems." Journal of Mathematical Analysis and Applications 475, no. 1 (2019): 385–402. http://dx.doi.org/10.1016/j.jmaa.2019.02.047.

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32

Bartušek, Miroslav. "On Structure of Solutions of a System of Four Differential Inequalities." gmj 2, no. 3 (1995): 225–36. http://dx.doi.org/10.1515/gmj.1995.225.

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Abstract The aim of the paper is to study a global structure of solutions of four differential inequalities with respect to their zeros. The structure of an oscillatory solution is described, and the number of points with trivial Cauchy conditions is investigated.
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33

He, Yang-Hui, Vishnu Jejjala, and Djordje Minic. "Eigenvalue Density, Li's Positivity, and the Critical Strip." inSTEMM Journal 1, S1 (2022): 1–14. http://dx.doi.org/10.56725/instemm.v1is1.23.

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We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution;this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann Xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach.
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34

Heath-Brown, D. R. "Zeros of systems of 𝔭-adic quadratic forms". Compositio Mathematica 146, № 2 (2010): 271–87. http://dx.doi.org/10.1112/s0010437x09004497.

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AbstractWe show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.
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35

Laurinčikas, Antanas. "On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function." Axioms 14, no. 3 (2025): 169. https://doi.org/10.3390/axioms14030169.

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The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function ζ(s) (zeros different from s=−2m, m∈N) lie on the critical line σ=1/2. In this paper, combining the universality property of ζ(s) with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts ζ(s+itτ) approximating the function ζ(s). Here, tτ denotes the Gram function, which is a continuous extension of the Gram points.
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36

Lagarias, Jeffrey C., and Brad Rodgers. "HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS." Quarterly Journal of Mathematics 71, no. 1 (2020): 257–80. http://dx.doi.org/10.1093/qmathj/haz043.

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Abstract The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.
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37

Šiaučiūnas, Darius, Raivydas Šimėnas, and Monika Tekorė. "Approximation of Analytic Functions by Shifts of Certain Compositions." Mathematics 9, no. 20 (2021): 2583. http://dx.doi.org/10.3390/math9202583.

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In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function.
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38

Foote, Richard, and V. Kumar Murty. "Zeros and poles of Artin L-series." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (1989): 5–11. http://dx.doi.org/10.1017/s0305004100001316.

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Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.
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39

O'MALLEY, ROBERT E. "NAIVE SINGULAR PERTURBATION THEORY." Mathematical Models and Methods in Applied Sciences 11, no. 01 (2001): 119–31. http://dx.doi.org/10.1142/s0218202501000787.

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The paper demonstrates, via extremely simple examples, the shocks, spikes, and initial layers that arise in solving certain singularly perturbed initial value problems for first-order ordinary differential equations. As examples from stability theory, they are basic to many asymptotic techniques. First, we note that limiting solutions of linear homogeneous equations [Formula: see text] on t≥0 are specified by the zeros of [Formula: see text], rather than by the turning points where a(t) becomes zero. Furthermore, solutions to the solvable equations [Formula: see text] for k=1, 2 or 3 can feature canards, where the trivial limit continues to apply after it becomes repulsive. Limiting solutions of the separable equation [Formula: see text] may likewise involve switchings between the zeros of c(x) located immediately above and below x(0), if they exist, at zeros of A(t). Finally, limiting solutions of many other problems follow by using asymptotic expansions for appropriate special functions. For example, solutions of [Formula: see text] can be given in terms of the Bessel functions Kj(t4/4ε) and Ij(t4/4ε) for j=3/8 and -5/8.
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40

Arias de Reyna, Juan, and Jan van de Lune. "On the exact location of the non-trivial zeros of Riemann's zeta function." Acta Arithmetica 163, no. 3 (2014): 215–45. http://dx.doi.org/10.4064/aa163-3-3.

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41

KELLIHER, JAMES P., and RIAD MASRI. "Analytic continuation of multiple Hurwitz zeta functions." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 3 (2008): 605–17. http://dx.doi.org/10.1017/s0305004107001028.

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AbstractWe use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to $\mathbb{C}^{d}$, to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of $\zeta_{d}(s;\theta)$.
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42

Laurinčikas, Antanas, and Darius Šiaučiūnas. "Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros." Mathematics 8, no. 11 (2020): 1936. http://dx.doi.org/10.3390/math8111936.

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Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any non-vanishing analytic function defined in the strip {s∈C:1/2<Res<1} with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞. Moreover, this set has a positive density for all but at most countably ε>0. The above approximation property remains valid for certain compositions F(ζ(s)).
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43

Guillory, Carroll. "A Characterization of a Sparse Blaschke Product." Canadian Mathematical Bulletin 32, no. 4 (1989): 385–90. http://dx.doi.org/10.4153/cmb-1989-056-0.

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AbstractWe give a characterization of a sparse Blaschke product b in terms of the separation of support sets of its zeros in M(H∞ + C) and the structure of the nonanalytic points. We use this characterization to give a sufficient condition on an interpolating Blaschke product q to have the following property: there exists a non trivial Gleason part P on which q is nonzero and less than one.
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44

Balčiūnas, Aidas, Virginija Garbaliauskienė, Julija Karaliūnaitė, Renata Macaitienė, Jurgita Petuškinaitė, and Audronė Rimkevičienė. "JOINT DISCRETE APPROXIMATION OF A PAIR OF ANALYTIC FUNCTIONS BY PERIODIC ZETA-FUNCTIONS." Mathematical Modelling and Analysis 25, no. 1 (2020): 71–87. http://dx.doi.org/10.3846/mma.2020.10450.

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In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.
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45

Alvarez, P. D. "Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice." Journal of Statistical Mechanics: Theory and Experiment 2024, no. 8 (2024): 083101. http://dx.doi.org/10.1088/1742-5468/ad64bc.

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Abstract We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.
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46

ODŽAK, ALMASA, and LEJLA SMAJLOVIĆ. "ON INTERPOLATION FUNCTIONS FOR GENERALIZED Li COEFFICIENTS IN THE SELBERG CLASS." International Journal of Number Theory 07, no. 03 (2011): 771–92. http://dx.doi.org/10.1142/s1793042111004356.

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We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class [Formula: see text] that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GL n(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function [Formula: see text].
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47

França, Guilherme, and André LeClair. "Some Riemann Hypotheses from random walks over primes." Communications in Contemporary Mathematics 20, no. 07 (2018): 1750085. http://dx.doi.org/10.1142/s0219199717500857.

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The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes, we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is [Formula: see text], then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this [Formula: see text] growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results, we obtain an equation which relates every individual non-trivial zero of the [Formula: see text]-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet [Formula: see text]-functions due to the existence of the pole at [Formula: see text], in which the Riemann [Formula: see text]-function is a particular case.
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48

Balčiūnas, Aidas, Violeta Franckevič, Virginija Garbaliauskienė, Renata Macaitienė, and Audronė Rimkevičienė. "UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION." Mathematical Modelling and Analysis 26, no. 1 (2021): 82–93. http://dx.doi.org/10.3846/mma.2021.12447.

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It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.
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49

CASTRO, CARLOS, and JORGE MAHECHA. "FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS." International Journal of Geometric Methods in Modern Physics 01, no. 06 (2004): 751–93. http://dx.doi.org/10.1142/s0219887804000393.

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The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert–Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet–Bandrauk–Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D=1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn=nπ and which coincide with the imaginary parts of the zeros of the function sinh (s). Finally, we discuss the relationship to the theory of 1/f noise.
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50

Laurinčikas, Antanas. "Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function." Studia Scientiarum Mathematicarum Hungarica 57, no. 2 (2020): 147–64. http://dx.doi.org/10.1556/012.2020.57.2.1460.

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AbstractLet 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.
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