Riemann Hypothesis Research Articles

ON THE LOWEST ZERO OF THE DEDEKIND ZETA FUNCTION

Abstract Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith.99(1) (2000), 61–65].

The variance of integers without small prime factors in short intervals

AbstractThe variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery’s Pair Correlation Conjecture and the Hardy–Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below y, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of y-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of y.

The Riemann Hypothesis and Exponentially Weighted Dirichlet Series

The Riemann Hypothesis and Exponentially Weighted Dirichlet Series

The Riemann Hypothesis and Polar Coordinates

The Riemann Hypothesis and Polar Coordinates

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along the primes, which is the natural pointwise convergence along prime result after Wierdl (1990) and a natural follow up after recent pointwise bilinear break-through by Krause Mirek-Tao (2021). In another project, the sharpest pointwise convergence average along primes for “near” integrable function is given, which is sharp up to Generalized Riemann Hypothesis.

A discrete mean value of the Riemann zeta function

AbstractIn this work, we estimate the sum over the nontrivial zeros of the Riemann zeta function where is a complex number with and and are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where is replaced by for all . 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 which was previously known under the generalized Riemann hypothesis, only in the case .

Some examples of well-behaved Beurling number systems

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called [ α , β ] [\alpha ,\beta ] -systems, where α \alpha and β \beta are connected to the optimal power saving in the prime and integer-counting functions. It is known that every [ α , β ] [\alpha ,\beta ] -system satisfies max { α , β } ≥ 1 / 2 \max \{\alpha ,\beta \}\ge 1/2 . In this paper we show there are [ α , β ] [\alpha ,\beta ] -systems for each α ∈ [ 0 , 1 ) \alpha \in [0,1) and β ∈ [ 1 / 2 , 1 ) \beta \in [1/2, 1) . Assuming the Riemann hypothesis, we also construct certain families of [ α , β ] [\alpha ,\beta ] -systems with β > 1 / 2 \beta >1/2 .

A generalization of a result of Báez-Duarte, Balazard, Landreau, and Saias

It is known that the Riemann hypothesis is true if and only if ▪ belongs to the Nyman-Beurling space, where ▪ denotes the characteristic function on the unit interval. Moreover, Báez-Duarte, Balazard, Landreau, and Saias provided a lower bound for the distance between ▪ and the subspaces of the Nyman-Beurling space. In this paper, we extend the result of Báez-Duarte et al. by considering more general characteristic functions. In addition, we give some applications and raise several questions related to the Riemann hypothesis.

Sign Changes of the Error Term in the Piltz Divisor Problem

Abstract We study the function $\Delta _{k}(x):=\sum _{n\leq x} d_{k}(n) - \mbox{Res}_{s=1} ( \zeta ^{k}(s) x^{s}/s )$, where $k\geq 3$ is an integer, $d_{k}(n)$ is the $k$-fold divisor function, and $\zeta (s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis (LH) is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon }$, such that $|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. In particular, $\Delta _{k}(x)$ does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $\Delta _{k}(x+h)-\Delta _{k}(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta _{k}(x)$, which we obtain by combining a method of Tsang with a technique of Lester.

Maximally elastic quadratic fields

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined assup⁡{sr:π1⋯πr=ρ1⋯ρs, with all πi,ρj irreducible}. We call a quadratic field Kmaximally elastic if the ring of integers of K is a UFD and each element of {1,32,2,52,3,…}∪{∞} 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.

The Riemann's Hypothesis, the Prime Numbers Theorem (PNT), and the Error

In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which allows us to limit the error with which said theorem predicts the value of the Prime-counting function π(x); and, in this way, endorse the veracity of the Riemann Hypothesis. Many people know that the Riemann Hypothesis is a difficult mathematical problem - even to understand - without a certain background in mathematics. Many techniques have been used, for more than 150 years, to try to solve it. Among them is the one that establishes that, if the Riemann hypothesis is true, then the error term that appears in the prime number theorem can be bounded in the best possible way. Specifically, Helge von Koch demonstrated in 1901 that it should be:

A smooth version of Landau’s explicit formula

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 [Formula: see text] is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order [Formula: see text]. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

On the Approximation of the Hardy Z-Function via High-Order Sections

The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, 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 ZN(t):=∑k=1Ncos(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)≈2ZN˜(t)(t) for N˜(t)=t2π. 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)≈ZN(t)(t) for N(t)=t2, 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.

Remarks on the Connection of the Riemann Hypothesis to Self-Approximation

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+iτ). 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.

A family of generalized Liénard differential systems with an explicit limit cycle

One of the important problems in the qualitative theory of real planar differential systems is limit cycles i.e., isolated periodic solutions in the set of all periodic this concept was defined at the end of the 19th century by Poincaré, and with the works of van der Pol and Liénard. The significance of these isolated solutions lies in their key role in understanding the dynamics of a certain differential system. Afterward, Hilbert specified a list of 23 problems for the advancement of mathematical science, and from then it started intensive research on these problems throughout the 20th century. Of the 23 problems, only the so-called 16th Hilbert’s problem and the Riemann conjecture remain open until now. The second part of the 16th Hilbert problem has two parts and asks for an upper bound on the number of possible limit cycles and their positions for a planar polynomial differential system of a given degree. The theory of the limit cycle operating as an indispensable mathematical tool has found broad and important applications in modern physics, chemical, biological, ,and other fields. Then the progress of these fields, in turn, promotes limit cycle research. In this paper, we prove the existence and uniqueness of the limit cycle for a family of generalized polynomial Liénard differential systems, the explicit expression of this limit cycle is given. An example illustrating our results is additionally presented.

Mass Equidistribution for Saito-Kurokawa Lifts

Let F be a holomorphic cuspidal Hecke eigenform for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{Sp}_{4}({\\mathbb{Z}})$\\end{document} 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.

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

ABSTRACT In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

Correlations of the Riemann zeta function

AbstractAssuming the Riemann hypothesis, we investigate the shifted moments of the zeta function introduced by Chandee Q. J. Math. 62(2011), no. 3, 545–572, where and satisfy and . We shall prove This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong [Can. J. Math. Published online 2023:1–31. DOI 10.4153/S0008414X23000548], particularly when the differences are unbounded as . The key insight is to combine work of Heap, Radziwiłł, and Soundararajan [Q. J. Math. 70 (2019), no. 4, 1387–1396] and work of the author [arXiv preprint arXiv:2301.10634 (2023)] with the work of Harper [arXiv preprint arXiv.1305.4618 (2013)] on the moments of the zeta function.

The Generalized Riemann Hypothesis from zeros of a single [formula omitted]-function

For each primitive Dirichlet character χ, a hypothesis GRH†[χ] is formulated in terms of zeros of the associated L-function L(s,χ). It is shown that for any such character, GRH†[χ] is equivalent to the Generalized Riemann Hypothesis.

A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation

It is true that my previous series of papers about the Riemann Hypothesis has raised lots of discussions or the concerns among the mathematics society. Indeed, some professionals may think that the Riemann Zeta function at s = 1 or ∑_(n=1)^∞▒1/n is actually divergent and may tend to an infinity, how it can be acted as a denominator in a proof or something like 1/∞ ? This author’s idea is there may be no need to evaluate such kind of summation as one may consider the sum just as an (imaginary) constant (but NOT the complex number). Or we may only consider such summation as a kind of improper integrals with a black-boxed answer. In other words, the true final value of the summation ∑_(n=1)^∞▒1/n is NOT an concerning serious issue. The function ∑_(n=1)^∞▒1/n^s is in practice a smooth function and equals to log N or the logarithmic function for s = 1 or have the infinite differentiable properties except n = 0, that is why the commercial software Maple Soft can compute the Zeta function’s associated Taylor series successfully. In fact, from the above series together with the harmonic properties, one may calculate the artifical Zeta function’s root model equation such as the 0.5 +/- β*cot(ln(x))/(αx+1)n (will be discussed in my next paper of the series). Indeed, this is NOT the best model. Actually, the true Riemann Zeta non-trivial Root model equation is: {z = x + yI | z∈0.5±(y_1⩽y⩽y_2 )I} where y1= (±cot(k))/ln(k) , y2 = (±tan(k))/ln(k) with 0<k<2π. Certainly, this author cannot avoid mistakes. Some of the defects in the previous paper such as one in the linguistic proof of the RH will be stated and amended. Lastly, this author wants to remind that the present series of paper is first to establish a model equation for zeta function’s roots and the Plank’s like constant. This author then applies the telescopic and logarithmic methods for a 3 cases proof of the RH. Also, we have a Matlab code for the evaluation of RH complex contour integral. Finally, the RH problem will be transformed into a linguistic one and hence we may have solved the Riemann Hypothesis issue completely through a 4 cases of truth table collaborations with the logical inference [7] & [15] to the causality (structural) organization of the sentences [16] or just named as “Logical & Organized Context”.