Riemann Hypothesis Research Articles

A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis

A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis

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.

Statistical Study of Euler’s Zeta Function by using Riemann Hypothesis

The aim of this paper to study the statistical measures for the zeta function and the relative comparison of the function. Here we give an examples of some of the suggested models and compare these models. The development of these models were done in a collection of machine learning algorithms for data mining tasks. As a measure of success of any model, we study the statistical measures CC, MAE, RMSE, RAE and RRSE. We would like to mention that there is no specific rational behind the choice of the statistical measures studied; except that they give good point-wise and overall statistics on the validity and success of the models. The statistical measures Root Mean Square Error and Mean Absolute Error are concerned with the average difference errors.

The Mathematical Heart of Tom Stoppard

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.

The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

AbstractOn 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 $a<b$ , 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 $0<{\gamma }\leq T$ .

Effective Sato–Tate conjecture for abelian varieties and applications

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 A' 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 A' have opposite sign.

On The Hidden Structure of Odd Numbers and Its Consequences for the Riemann Hypothesis

The Riemann hypothesis remains unconfirmed or invalidated to this day, although local verifications on the calculation of its zeros have never found it faulty. Mertens reformulated the problem to make it much more accessible and surely more easily solvable. Unfortunately, his conjecture, also called a strong conjecture, turned out to be incorrect. There remain 2 other conjectures, the weak and the general, which do not yet have fixed status. Would it then be possible that the resolution of the Riemann hypothesis arises through one of these 2 conjectures? We answer yes, and we turn our attention to Mertens' weak conjecture. Equipped with a new equation to date and a methodical approach that uses a bounded description of numbers, we solve the conjecture by placing ourselves under the criteria of Hausdorff's theorem concerning the evolution of the sum, by showing first of all that odd numbers have a structure similar to that of triangular numbers, and then the randomness arises from their intrinsic regularity; which does not contradict the Martin-L\"{o}f definition of random sequences despite everything. We therefore resolve the Riemann hypothesis and we provide an equation which will certainly make it possible to resolve other types of problems, and thus to extend the means made available to mathematicians to examine various types of questions whether in number theory or in other fields of mathematics, or even in physics, cryptography and computer science.

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.

An Extension Proof of Riemann Hypothesis by a Logical Entails Truth Table

There were many mathematicians who tried to prove or disprove the statement of Riemann Hypothe-sis. However, none of them have been successfully approved by the Clay Mathematical Institute. In addition, to the best of this author’s knowledge, these mathematicians haven’t employed the technique of logical truth table during their proofs. With reference to this author’s previous proof in [1], this author have employed the method of multiplicative telescope together with the prime boundary gaps. In this extended version of my proof to the Riemann Hypothesis, this author tries to show that RH statement is true through the four cases of the conditional statements in the truth table. Three of the cases (I, II, IV) are found to be true for the conditional statement in the Riemann Hypothesis while only one (case III) is found to be false (and acts as the disproof by a counter-example). Moreover, there are also three sub-cases (i, ii, iii) [1] among these four tabled cases. The main idea is that the we may disproof the hypothesis statement that is similar to the RH one by first find a counter-example which is obviously a disproof (case III) to the (Riemann) hypothesis. But it is NOT compatible with the GÖdel’s Incompleteness Theorem. Otherwise either the disproof to the statement or the Gödel is incorrect which is impossible. Hence, the disproof is said to be incompatible with the Gödel. On the other hand, all of the other truth cases (I, II, IV) for the statement are indeed the examples for the pos-itive results to the Riemann Hypothesis statement and are compatible with the Gödel. Therefore, the only way to make a conclusion is to say or force the Riemann Hypothesis statement to be correct.In general, for any hypothesis with the conditional statements structure like the Riemann one, we may also prove them by the similar techniqe and the arguments of the truth table for their conditional statements together with the Gödel’s Incompleteness theorem to force the positive result for the hy-pothesis statement. Actually, there are many applications for the truth tables especially in the fields like language (structure & modeling) or in engineering (logic gates & programming) etc during our everyday usage.

On the Primorial Counting Function

Let Nk denote the kth primorial and let K(x) be the primorial counting function, the function that counts the number of primorials not exceeding x. In this note, we examine the sign of the functions K ( x ) − π ( log x ) and K ( x ) − li ( log x ) and find a new necessary and sufficient criterion for the truth of the Riemann hypothesis.

On the Lindelöf hypothesis for general sequences

AbstractIn a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

Weighted central limit theorems for central values of $L$-functions

We establish a central limit theorem for the central values of Dirichlet L -functions with respect to a weighted measure on the set of primitive characters modulo q as q \rightarrow \infty . Under the Generalized Riemann Hypothesis (GRH), we also prove a weighted central limit theorem for the joint distribution of the central L -values corresponding to twists of two distinct primitive Hecke eigenforms. As applications, we obtain (under GRH) positive proportions of twists for which the central L -values simultaneously grow or shrink with q as well as a positive proportion of twists for which linear combinations of the central L -values are non-zero.

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.

BOUNDING ZETA ON THE 1-LINE UNDER THE PARTIAL RIEMANN HYPOTHESIS

AbstractWe 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.

Negative moments of the Riemann zeta-function

Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in ζ ⁢ ( s ) {\zeta(s)} . For example, integrating | ζ ⁢ ( 1 2 + α + i ⁢ t ) | - 2 ⁢ k {|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}} with respect to t from T to 2 ⁢ T {2T} , we obtain an asymptotic formula when the shift α is roughly bigger than 1 log ⁡ T {\frac{1}{\log T}} and k < 1 2 {k<\frac{1}{2}} . We also obtain non-trivial upper bounds for much smaller shifts, as long as log ⁡ 1 α ≪ log ⁡ log ⁡ T {\log\frac{1}{\alpha}\ll\log\log T} . This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function.

Behavior of Non-trivial Zeros and Prime Numbers in Riemann Hypothesis

Behavior of Non-trivial Zeros and Prime Numbers in Riemann Hypothesis

Character sums and the Riemann Hypothesis

Character sums and the Riemann Hypothesis

Quantum Hidden Variables and Riemann Hypothesis

Quantum Hidden Variables and Riemann Hypothesis

Riemann Hypothesis Disproved - New Complex Zeros of Zeta Function Found off the Critical Line in the Critical Strip

Riemann Hypothesis Disproved - New Complex Zeros of Zeta Function Found off the Critical Line in the Critical Strip

An Extension Proof of Riemann Hypothesis by a Logical Entails Truth Table

An Extension Proof of Riemann Hypothesis by a Logical Entails Truth Table