Generalized Riemann Hypothesis Research Articles

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis

An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis

On the Universal Presence of Mathematics for Incompletely Predictable Problems in Rigorous Proofs for Riemann Hypothesis, Modified Polignac’s and Twin Prime Conjectures

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

A note on the two variable Artin's conjecture

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, mamodp∈〈b〉modp} 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, mamodp∈〈b〉modp}≫xlog2⁡x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S={m1,m2,m3} such thatm1,m2,m3,−3m1m2,−3m2m3,−3m1m3,m1m2m3 are not squares, there exists a pair of elements a,b∈S such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x. Further, under the assumption of a level of distribution greater than x23 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={m1,m2} such thatm1,m2,−3m1m2 are not squares, there exists a pair of elements a,b∈{m1,m2,−3m1m2} such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x.

Discrete mean values of the product of derivatives of shifted Dirichlet L-functions

Assuming the Generalized Riemann Hypothesis, we calculate ∑0<γχ≤TL(μ)(ρχ+iδ,χ)L(ν)(1−ρχ−iδ,χ‾), which extends the result proposed by Gonek in 1984.

Riemann zeros as quantized energies of scattering with impurities

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 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document} 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 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}. 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 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{D}\\left(s\\right)$$\\end{document}, 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 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

Higher-degree Artin conjecture

ABSTRACT 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.

On the effective version of Serre's open image theorem

AbstractLet be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod Galois representation of is surjective for each prime number that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime , linear in the logarithm of the conductor of , such that is nonsurjective.

Beyond the Riemann Hypothesis [v2

The functional equation of real variable that Riemann used in his paper was subjected to elementary operations. And I obtained a lot of complex functional equations that the Riemann zeta function follows respectively. Here, functional equation transformations were the main methods for obtaining the complex functional equations. Half of those are equivalent to the complete symmetric functional equation that the Riemann Xi function follows, and one of those has an origin symmetry with correction terms. From the origin symmetric functional equation including correction terms, the representation containing the leading term of the zeta function for any complex number was obtained. And the Riemann hypothesis was proved by applying reduction to absurdity. Moreover the general representation containing the leading term of the zeta function for any odd number of 3 or more was also obtained. By suitably combining those functional equations, I observed a new explicit formula for the zeta function. The Riemann hypothesis was again proven using the deductive method. And two types of general representations for the zeta function for any odd number of either 3 or 7, or more, were also obtained from the explicit formula. In total, three types of general representations for the zeta function for any odd number of either 3 or 7, or more, were discovered. Conversely, I defined a new function, named the Chi function, for the left side of the origin symmetric functional equation that includes corrective terms. The Chi function is similar to the Riemann Xi function and exhibits origin symmetry. Furthermore, I defined a new function, the eta function, which is similar to the zeta function. The eta function’s pole and trivial zeros are the same as those of the zeta function. Furthermore, the Chi and eta functions have the same non-trivial zeros on the imaginary axis. And I proposed a generalized Riemann hypothesis for the eta function that states that all non-trivial zeros lie on the imaginary axis. Since I was able to discover the explicit formula for the eta function, the deductive method was used to prove the generalized Riemann hypothesis for the eta function. As you know, there are different types of transformations between the prime numbers and the non-trivial zeros of the zeta function. I discovered that there are comparable transformations between the prime numbers and the non-trivial zeros of the eta function. Based on the results of numerical experiments, I proposed some conjectures referring to relationships between the prime numbers and non-trivial zeros of the eta function.

Mollified moments of cubic Dirichlet L-functions over the Eisenstein field

We prove, assuming the generalized Riemann Hypothesis (GRH) that there is a positive density of L-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point s=1/2. This is achieved by computing the first mollified moment, which is obtained unconditionally, and finding a sharp upper bound for the higher mollified moments for these L-functions, under GRH. The proportion of non-vanishing is explicit, but extremely small.

Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions

ABSTRACT In this paper, we provide Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet L-functions. The details are as follows. Let L(s, χ) be the Dirichlet L-function and G(χ) be the Gauss sum associated with a primitive Dirichlet character χ (mod q). We define f ( s , χ ) : = q s L ( s , χ ) + i − κ ( χ ) G ( χ ) L ( s , χ ¯ ) , where χ ¯ is the complex conjugate of χ and κ(χ) := (1 – χ(−1))/2. Then, we prove that f (s, χ) satisfies Riemann’s functional equation in Hamburger’s theorem if χ is even. In addition, we show that f (σ, χ) ≠ 0 for all σ ≥ 1. Moreover, we prove that f(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1 if and only if L(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1. When χ is real, all zeros of f (s, χ) with ℜ ( s ) &gt; 0 are on the line σ = 1/2 if and only if the generalized Riemann hypothesis for L(s, χ) is true. However, f (s, χ) has infinitely many zeros off the critical line σ = 1/2 if χ is non-real.

Ratios conjecture for quadratic Hecke L-functions in the Gaussian field

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke L-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of L-functions, obtaining an error term of size O(X1/2+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(X^{1/2+\\varepsilon })$$\\end{document}.

Two problems on the distribution of Carmichael's lambda function

AbstractLet denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by as n varies. Under the stronger assumption that , we prove that equidistribution persists throughout a Siegel–Walfisz‐type range of uniformity. By similar methods we show that obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of a mod n, for any fixed integer .

Distinguishing newforms by the prime divisors of their Fourier coefficients

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis.

Extreme values of derivatives of zeta and L-functions.

It is proved that as , uniformly for all positive integers , we have where. Here, is the Dickman function. We have and when , which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for and . Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where is prime and isgiven.

Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \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 Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | &gt; 10 4 |d| &gt; 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( − 2289 ) \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 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest.

The prime number theorem for primes in arithmetic progressions at large values

Abstract Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet L-functions is true, we then establish explicit formulae for $\psi(x,\chi)$, $\theta(x,\chi)$ and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.

Conditional estimates for the logarithmic derivative of Dirichlet [formula omitted]-functions

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the q-aspect for the logarithmic derivative L′/Lσ,χ of Dirichlet L-functions, where χ is a primitive character modulo q≥1030 and 1/2+1/loglogq≤σ≤1−1/loglogq. In addition, for σ=1 we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.

Poisson Behavior for Chains of Small Quadratic Non-Residues

Abstract It is known that under the Generalized Riemann Hypothesis, the smallest quadratic non-residue modulo a prime $p$ is less than or equal to $(\log p)^{2}$. In ranges slightly larger, of size $(\log p)^{A}$ with $A&amp;gt;2$, we consider chains of $r$ consecutive quadratic non-residues. We prove unconditionally that for almost all primes $p$ in short intervals, these chains exhibit Poisson behavior.

Moments of moments of primes in arithmetic progressions

AbstractWe establish unconditional ‐results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under the Generalized Riemann Hypothesis (GRH). Finally, under GRH and the Linear Independence Hypothesis (LI), we prove an asymptotic for all moments of the associated limiting distribution, which, in turn, indicates that our unconditional and GRH results are essentially best possible. Using our probabilistic results, we formulate a conjecture on the moments with a precise associated range of validity, which we believe is also best possible. This last conjecture implies a ‐analog of the Montgomery‐Soundararajan conjecture on the Gaussian distribution of primes in short intervals. The ideas in our proofs include a novel application of positivity in the explicit formula and the combinatorics of arrays of characters that are fixed by certain involutions.