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The International Conference on Mathematical Analysis, Applications and Computational Simulation (ICMAACS) is discussed in this article, along with the associated awards. The conference serves as a platform for mathematicians and scientists to share advancements in pure and applied mathematics. The Qin-Jiu-Shao prize is given to mathematicians who have made exceptional contributions, while the Yang-Hui award recognizes those who have produced internationally outstanding works. The article also mentions the winners of these awards for the 2023 conference, as well as the recipients of the ICMAACS Best Paper Presentation Award, which includes younger mathematicians and scientists from various countries. Additionally, the text highlights the contributions of Professor Torres and coauthors in the field of COVID-19 modeling and dynamic control. [Extracted from the article]

We provide explicit bounds for the Riemann zeta-function on the line $\mathrm {Re}\,{s}=1$ , assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Let F be a holomorphic cuspidal Hecke eigenform for of weight k that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of F equidistributes on the Siegel modular variety as k⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity. [ABSTRACT FROM AUTHOR]

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$ th moments of the Riemann zeta function on the critical line. [ABSTRACT FROM AUTHOR]

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$ -function associated with the primitive Maass cusp form over $\Gamma _0(N)$. [ABSTRACT FROM AUTHOR]

In this work, we estimate the sum ∑0<ℑ(ρ)⩽Tζ(ρ+α)X(ρ)Y(1−ρ)$$\begin{align*} \sum _{0 < \Im (\rho) \leqslant T} \zeta (\rho +\alpha)X(\rho) Y(1\!-\! \rho) \end{align*}$$over the nontrivial zeros ρ$\rho$ of the Riemann zeta function where α$\alpha$ is a complex number with α≪1/logT$\alpha \ll 1/\log T$ and X(·)$X(\cdot)$ and Y(·)$Y(\cdot)$ are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where ζ(ρ+α)$\zeta (\rho +\alpha)$ is replaced by ζ(m)(ρ)$\zeta ^{(m)}(\rho)$ for all m∈N$m\in \mathbb {N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann hypothesis, we obtain the lower bound ∑0<ℑ(ρ)

Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function Mα,β(T)=∫T2T∏k=1mζ12+i(t+αk)2βkdt$$\begin{equation*} \hspace*{24.5pt}M_{{\bm \alpha}, {\bm \beta}} (T) = \int _T^{2T} \prod _{k = 1}^m {\left|\zeta \left(\tfrac{1}{2} + i (t + \alpha _k)\right)\right|}^{2 \beta _k} dt\hspace*{-24.5pt} \end{equation*}$$introduced by Chandee Q. J. Math. 62(2011), no. 3, 545–572, where α=α(T)=(α1,...,αm)${\bm \alpha} = {\bm \alpha} (T) = (\alpha _1, \ldots, \alpha _m)$ and β=(β1...,βm)$\bm {\beta } = (\beta _1 \ldots, \beta _m)$ satisfy |αk|⩽T/2$|\alpha _k| \leqslant T/2$ and βk⩾0$\beta _k\geqslant 0$. We shall prove Mα,β(T)≪βT(logT)β12+⋯+βm2∏1⩽j

For a real number k, define π k (x) = ∑ p ≤ x p k . When k > 0 , we prove that π k (x) - π (x k + 1) = Ω ± x 1 2 + k log x log log log x as x → ∞ , and we prove a similar result when - 1 < k < 0 . This strengthens a result in a paper by Gerard and the author and it corrects a flaw in a proof in that paper. We also quantify the observation from that paper that π k (x) - π (x k + 1) is usually negative when k > 0 and usually positive when - 1 < k < 0 . [ABSTRACT FROM AUTHOR]

The Z-function is the real function given by Z (t) = e i θ (t) ζ 1 2 + i t , where ζ (s) is the Riemann zeta function, and θ (t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z (t) and its zeros. The sections of Z (t) are given by Z N (t) : = ∑ k = 1 N cos (θ (t) − ln (k) t) k for any N ∈ N . Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z (t) ≈ 2 Z N ˜ (t) (t) for N ˜ (t) = t 2 π . While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z (t) ≈ Z N (t) (t) for N (t) = t 2 , which is Spira's approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira's conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. [ABSTRACT FROM AUTHOR]

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n, d n ln (f (z)) d z n = d n ln (g (z)) d z n . The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic θ 1 functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions. [ABSTRACT FROM AUTHOR]

We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on Γ 0 (N) are on the circle | z | = 1 ∕ N under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

In this note, we study the Li coefficients $\lambda _{n,a}$ for the quadrilateral zeta function. Furthermore, we give an arithmetic and asymptotic formula for these coefficients. Especially, we show that for any fixed $n \in {\mathbb {N}}$ , there exists $a>0$ such that $\lambda _{2n-1,a}> 0$ and $\lambda _{2n,a}. [ABSTRACT FROM AUTHOR]

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined as sup ⁡ { s r : π 1 ⋯ π r = ρ 1 ⋯ ρ s , with all π i , ρ j irreducible }. We call a quadratic field K maximally elastic if the ring of integers of K is a UFD and each element of { 1 , 3 2 , 2 , 5 2 , 3 , ... } ∪ { ∞ } appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that K = Q (2) is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields. [ABSTRACT FROM AUTHOR]

Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery's function F(\alpha, T) over long intervals by means of a Fourier optimization framework. The function F(\alpha, T) is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach: (i) the introduction of new averaging mechanisms in our conceptual framework and (ii) the full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions. We conclude that such an average value, that is conjectured to be 1, lies between 0.9303 and 1.3208. Our Fourier optimization framework also yields an improvement on the current bounds for the analogous problem concerning the non-trivial zeros in the family of Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic ranks of the Hasse–Weil L-function. We consider the products for the completed Hasse–Weil L-function in the framework of the theory of modular forms. We propose the new conjectures for the deformed Fourier sine and cosine integral formulas related to the completed Hasse–Weil L-functions and Mittag-Leffler function. We show the entire-function solutions for the fractional diffusion equations in mathematical physics. We also discuss the trivial zeros of the Hasse–Weil L-function. [ABSTRACT FROM AUTHOR]

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number μ is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order μlog3 2μ. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished. [ABSTRACT FROM AUTHOR]

Several methods are used to evaluate finite trigonometric sums. In most cases, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We establish both reciprocity and three sum relations for trigonometric sums. Motivated by certain sums that we have evaluated, we add coprime conditions to the summands and thereby define analogues of Ramanujan sums, which we in turn evaluate. One of these analogues leads to a criterion for the Riemann Hypothesis, analogous to the Franel–Landau criterion. [ABSTRACT FROM AUTHOR]

This paper gives an explicit version of Selberg's mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32\,277\log ^2 y]$ for at least half the $y\in [x,2x]$ for all $x\geq 2$ , and at least one Goldbach number in $(x,x+9696 \log ^2 x]$ for all $x\geq 2$. [ABSTRACT FROM AUTHOR]

Assuming the Generalized Riemann Hypothesis, we calculate ∑ 0 < γ χ ≤ T L (μ) (ρ χ + i δ , χ) L (ν) (1 − ρ χ − i δ , χ ‾) , which extends the result proposed by Gonek in 1984. [ABSTRACT FROM AUTHOR]

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ (s) by shifts ζ (s + i τ) . In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ (s) with all but at most countably many accuracies ε > 0 . Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. [ABSTRACT FROM AUTHOR]

For E1$E_1$ and E2$E_2$ elliptic curves defined over a number field K$K$, without complex multiplication, we consider the function FE1,E2(x)${\mathcal {F}}_{E_1, E_2}(x)$ counting nonzero prime ideals p$\mathfrak {p}$ of the ring of integers of K$K$, of good reduction for E1$E_1$ and E2$E_2$, of norm at most x$x$, and for which the Frobenius fields Q(πp(E1))$\mathbb {Q}(\pi _{\mathfrak {p}}(E_1))$ and Q(πp(E2))$\mathbb {Q}(\pi _{\mathfrak {p}}(E_2))$ are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that E1$E_1$ and E2$E_2$ are not potentially isogenous if and only if FE1,E2(x)=oxlogx${\mathcal {F}}_{E_1, E_2}(x) = \operatorname{o}\left(\frac{x}{\operatorname{log}x}\right)$, we investigate the growth in x$x$ of FE1,E2(x)${\mathcal {F}}_{E_1, E_2}(x)$. We prove that if E1$E_1$ and E2$E_2$ are not potentially isogenous, then there exist positive constants κ(E1,E2,K)$\kappa (E_1, E_2, K)$, κ′(E1,E2,K)$\kappa ^{\prime }(E_1, E_2, K)$, and κ′′(E1,E2,K)$\kappa ^{\prime \prime }(E_1, E_2, K)$ such that the following bounds hold: (i) FE1,E2(x)<κ(E1,E2,K)x(loglogx)19(logx)1918${\mathcal {F}}_{E_1, E_2}(x) < \kappa (E_1, E_2, K) \frac{ x (\operatorname{log}\operatorname{log}x)^{\frac{1}{9}}}{ (\operatorname{log}x)^{\frac{19}{18}}}$; (ii) FE1,E2(x)<κ′(E1,E2,K)x67(logx)57${\mathcal {F}}_{E_1, E_2}(x) < \kappa ^{\prime }(E_1, E_2, K) \frac{ x^{\frac{6}{7}}}{ (\operatorname{log}x)^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) FE1,E2(x)<κ′′(E1,E2,K)x23(logx)13${\mathcal {F}}_{E_1, E_2}(x) < \kappa ^{\prime \prime }(E_1, E_2, K) x^{\frac{2}{3}} (\operatorname{log}x)^{\frac{1}{3}}$ under GRH, Artin's Holomorphy Conjecture for the Artin L$L$‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin L$L$‐functions of number field extensions. [ABSTRACT FROM AUTHOR]

Let (n, q) denote the greatest common divisor of positive integers n and q, and let f r denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full ( r = 2 ) and cube-full numbers ( r = 3 ), which is ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f r q d log x q with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with r = 2 under the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

The text is a summary of a poetry collection titled "Mathematical Graffiti: Bridges 2023 Clerihew Collection," curated by E.R. Lutken. The collection features clerihew poems about well-known mathematicians throughout history. Clerihews are biographical and humorous poems consisting of two couplets with a rhyming pattern of aabb. The collection includes poems about mathematicians such as Pythagoras, Euclid, Archimedes, Hypatia of Alexandria, Leonardo Fibonacci, Tycho Brahe, John Napier, and many others. The poems explore various mathematical concepts and historical figures in a whimsical and entertaining way. [Extracted from the article]

This article presents a concise proof of the famous Benford's law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of human number system. The criterion can bring great convenience to the field of fraud detection. [ABSTRACT FROM AUTHOR]

In this paper, we find some conditions under which a sequence \{\alpha (n)\} will satisfy the Laguerre inequality of any order asymptotically. Using this method, we prove that for any r and some constant c, the Maclaurin coefficients \gamma (n) of the Riemann Xi-function satisfy the Laguerre inequality of order r when n>cr^3, which provides a necessary condition for the Riemann hypothesis. We also prove that the partition function satisfies the Laguerre inequality of order r\geq 5 when n\geq 6r^4. As a consequence, it gives an affirmative answer to Wagner's conjecture on the threshold for the Laguerre inequalities of order no more than 10 for the partition function. Moreover, motivated by the study of Craven and Csordas on the complete monotonicity of the Maclaurin coefficients of entire functions in Laguerre-Pólya class, we consider the complete monotonicity of the sequences \{\alpha (n)\}. We give the criteria for the asymptotically complete monotonicity of the sequence \{\alpha (n)\} and \{\log \alpha (n)\}, respectively. With this criteria, we show that (-1)^r \Delta ^r \gamma (n)>0 for n>ce^{r^3} and (-1)^{r-1} \Delta ^r \log \gamma (n)>0 for n>cr^2. Furthermore, we propose some open problems. [ABSTRACT FROM AUTHOR]

Let S(t)=1πImlogζ12+it$S(t) = \frac{1}{\pi }\operatorname{Im}\log \zeta \left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets {t∈[T,2T]:S(t)⩾V}$\lbrace t\in [T,2T] \colon S(t) \geqslant V\rbrace$ for loglogT⩽V≪logTloglogT1/3$\sqrt {\log \log T} \leqslant V \ll \left(\frac{\log T}{\log \log T}\right)^{1/3}$. For V⩽(logT)1/3−ε$V \leqslant (\log T)^{1/3-\varepsilon }$, our bound has a Gaussian shape with variance proportional to loglogT$\log \log T$. At the endpoint, V≍logTloglogT1/3$V \asymp \left(\frac{\log T}{\log \log T}\right)^{1/3}$, our result implies the best known Ω$\Omega$‐theorem for S(t)$S(t)$ that is due to Tsang. We also explain how the method breaks down for V≫logTloglogT1/3$V \gg \left(\frac{\log T}{\log \log T}\right)^{1/3}$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range loglogT⩽V≪logTloglogT1/2$\sqrt {\log \log T} \leqslant V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$. [ABSTRACT FROM AUTHOR]

In this paper, we prove Selberg's announced result on r$r$‐gaps between zeros of the Riemann zeta‐function ζ$\zeta$. Our proof uses a result on variations of argζ$\arg \zeta$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage‐Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments. [ABSTRACT FROM AUTHOR]

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin's primitive root conjecture that the order is 'almost always almost maximal' assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments. [ABSTRACT FROM AUTHOR]

Let f be a newform for the group |$\Gamma_{0}(q)$| and χ d be a primitive quadratic character of conductor |$|d|$|⁠. In this article, we prove an asymptotic for the second moment of the derivative of |$L(s,\, f \otimes \chi_{8d})$| at the central point |$1/2$|⁠ , which was previously known under grand Riemann hypothesis (GRH) by Petrow [ 9 ]. [ABSTRACT FROM AUTHOR]

This work is an attempt to apply Lambert series in the theory of univalent functions. We first consider the Hadamard product of Rabotnov function and Lambert series with coefficients derived from the arithmetic function σ n to introduce a normalized linear operator J R α , β z . We then acquire sufficient conditions for J R α , β z to be univalent, starlike and convex, respectively. Furthermore, we discuss the inclusion results in some special classes, namely, spiral-like and convex spiral-like subclasses. In addition, we extend the findings by incorporating two Robin's inequalities, one of which is analogous to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }})^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$. The same result holds when the ${\gamma }$ 's are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0. [ABSTRACT FROM AUTHOR]

The purpose of this note is to show why and how the fractional operators required for formalizing dynamic issues in complex environment, mobilize among the most advanced mathematical concepts : topos, site, spectrum, sheaf, ringed spaces, p-adic numbers, etc. In the context of physical exchanges in spaces crumpled by long range interaction (hyperbolic geometries), geodesics no longer respond to Noether invariance principles but to memory effects and to the categorical limits imposed by some requirements for completion. This completion is in fine based upon algebra-topology coupling. This coupling may be expressed through the zeta function which, as a bridge, ensures the formal closure of the representation and implicitly the space-time characteristics. The analysis points out the existence of a particular case corresponding to the Riemann hypothesis (and Martin’s axiom). Above approach involves a relationship between quantum mechanics and 2D self-similarity. In non integer cases, time is discretized and an arrow of time naturally emerges from the analysis. The note relates this arrow to anti entropic properties of dissipative complex systems. [ABSTRACT FROM AUTHOR]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse--Weil bound. Third, for a pair of abelian varieties A and A0 defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A0 have opposite sign. [ABSTRACT FROM AUTHOR]

The article informs about new effective results in the theory of the Riemann zeta-function, focusing on four groups providing estimates for the zeta-function and associated functions under the assumption of the Riemann hypothesis. Topic include explicit and RH estimates for various functions related to the zeta-function, including their applications to the distribution of prime numbers and other arithmetic properties, emphasizing the importance of these findings in mathematical research.

For a natural number n , let Z 1 (n) : = ∑ ρ n ρ ρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q ≥ 3 , we define Z 1 (s , χ) : = ∑ n = 1 ∞ χ (n) Z 1 (n) n s for ℜ (s) > 2 and obtain the meromorphic continuation of the function Z 1 (s , χ) to the region ℜ (s) > 1 2 . Our main result indicates that the poles of Z 1 (s , χ) in the region 1 2 < ℜ (s) < 1 , if they exist, are related to the zeros of many Dirichlet L -functions in the same region. [ABSTRACT FROM AUTHOR]

Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M /ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results. [ABSTRACT FROM AUTHOR]

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ(s) function along the axis of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis . The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function , i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis , there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a Post-Widder-type inversion formula for a Widder–Lambert-type integral transform. We also obtain an $ L^p $ L p inversion formula, $ 1 \lt p\leq ~2 $ 1 < p ≤ 2 , for this transform by means of the Mellin transform. As a consequence of these results, we consider some class of functions regarding the Salem's equivalence to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

Let E/Q$E/\mathbb {Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod ℓ$\ell$ Galois representation ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ of E$E$ is surjective for each prime number ℓ$\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime ℓ$\ell$, linear in the logarithm of the conductor of E$E$, such that ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ is nonsurjective. [ABSTRACT FROM AUTHOR]

"An Introduction to the Circle Method" by M. Ram Murty and Kaneenika Sinha is a book that introduces the circle method in number theory. The authors assume no prior background in number theory but do expect some familiarity with complex analysis. The book provides a detailed explanation of the circle method, including the use of exponential sums and contour integration. It also covers the steps that are often omitted in other texts, making it a valuable resource for both beginners and experts in the field. The book focuses on building a narrow path towards solving Waring's Problem and Goldbach's Conjecture in additive number theory. The circle method has been successful in proving the Ternary Goldbach Theorem, but has not been able to solve Goldbach's Conjecture. The authors argue that the obstruction to solving Goldbach's Conjecture is more serious than previously thought. The book acknowledges the uncertainty of the future of the circle method due to ongoing developments in analytic number theory. Overall, the book is praised for making the circle method accessible to a wider audience. [Extracted from the article]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over \mathbb {Q}. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields \mathbb {Q}(\sqrt {d}) with |d| < 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields, including \mathbb {Q}(\sqrt {213}) and \mathbb {Q}(\sqrt {-2289}). To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X_0(125) and X_0(169), which may be of independent interest. [ABSTRACT FROM AUTHOR]

An explicit subconvexity bound for the Riemann zeta function ζ (s) on the critical line s = 1 / 2 + i t is proved. This bound uses a new version of the Kusmin–Landau lemma and improves on current explicit bounds for | ζ (1 / 2 + i t) |. [ABSTRACT FROM AUTHOR]

The Riemann hypothesis (RH) is one of the major unsolved problems in pure mathematics. In the present paper, a parameterized family of non-cooperative games is constructed with the property that, if RH is true, then any game in the family admits a unique Nash equilibrium. We argue that this result is not degenerate. Indeed, neither is the conclusion a tautology, nor is RH used to define the family of games. • The Riemann Hypothesis (RH) is a famous open problem in mathematics. • This paper constructs generalized difference-form contests related to RH. • These contests admit a unique Nash equilibrium if RH holds true. • We argue that the result is not degenerate in any obvious way. [ABSTRACT FROM AUTHOR]

Let k be a number field and G be a finite group. Let F k G (Q) be the family of number fields K with absolute discriminant D K at most Q such that K / k is normal with Galois group isomorphic to G . If G is the symmetric group S n or any transitive group of prime degree, then we unconditionally prove that for all K ∈ F k G (Q) with at most O ε (Q ε) exceptions, the L -functions associated to the faithful Artin representations of Gal (K / k) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree n S n -fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; for a prime p , the periodic torus orbits attached to the ideal classes of almost all totally real degree p fields F over ℚ equidistribute on PGL p (Z) ∖ PGL p (R) with respect to Haar measure; for each ℓ ≥ 2 , the ℓ -torsion subgroups of the ideal class groups of almost all degree p fields over k (resp. almost all degree n S n -fields over k ) are as small as GRH implies; and an effective variant of the Chebotarev density theorem holds for almost all fields in such families. [ABSTRACT FROM AUTHOR]

Mathematics represents a conspicuous through-line in Tom Stoppard's body of work. The allusions to the Riemann hypothesis in the recently produced Leopoldstadt (2020) form a notable bookend with the musings on probability in Rosencrantz & Guildenstern Are Dead (1966) that launched Stoppard's career. In the intervening decades, the playwright's ongoing self-education in mathematics surfaces in a series of cleverly staged debates about providence versus free will, the existence of moral absolutes, and the origins of human consciousness. Contrary to expectations, the mathematics in these plays most often serves as a catalyst for the persistence of uncertainty in a rising tide of rationalism. The decision to place a pure mathematician at the centre of Leopoldstadt, an autobiographically-inspired family epic about the Holocaust, reveals the depth of Stoppard's fondness for mathematics and his faith in its potential to tell human stories. As it has in earlier plays, the mathematics in Leopoldstadt offers a distinctive commentary about the tension between rational order and randomness while also making a quiet case for the perseverance of beauty in the darkest of possible worlds. [ABSTRACT FROM AUTHOR]