Reduced Residue Classes Research Articles

The Prime Geodesic Theorem in Arithmetic Progressions

Abstract We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

The Barban–Davenport–Halberstam theorem for a restricted set of arithmetic progressions

We consider the asymptotic version of the Barban–Davenport–Halberstam theorem in the form proved first by Hooley. Instead of averaging the expected error term over all reduced residue classes to modules in a given range, here we only consider those reduced arithmetic progressions that satisfy an additional congruence condition and provide an asymptotic approximation.

On a conjecture of Soundararajan

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,\chi)$ for characters $\chi$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,\chi)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.

On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

The smallest invariant factor of the multiplicative group

Let [Formula: see text] denote the least invariant factor in the invariant factor decomposition of the multiplicative group [Formula: see text]. We give an asymptotic formula, with order of magnitude [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. We also give an asymptotic formula, for any even [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. These results require a version of the Selberg–Delange method whose dependence on certain parameters is made explicit, which we provide in Appendix A. As an application, we give an asymptotic formula for the counting function of those integers [Formula: see text] all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

Generalized Riemann hypothesis, time series and normal distributions**Dedicated to Giorgio Parisi on the occasion of his 70th birthday.

L functions based on Dirichlet characters are natural generalizations of the Riemann function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the generalized Riemann hypothesis relative to the non-trivial complex zeros of the Dirichlet L functions by studying the possibility to enlarge the original domain of convergence of their Euler product. The feasibility of this analytic continuation is ruled by the asymptotic behavior in N of the series involving Dirichlet characters modulo q on primes pn. Although deterministic, these series have pronounced stochastic features which make them analogous to random time series. We show that the BN’s satisfy various normal law probability distributions. The study of their large asymptotic behavior poses an interesting problem of statistical physics equivalent to the single Brownian trajectory problem, here addressed by defining an appropriate ensemble involving intervals of primes. For non-principal characters, we show that the series BN present a universal diffusive random walk behavior in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo q and the Lemke Oliver–Soundararajan conjecture on the distribution of pairs of residues on consecutive primes. This purely diffusive behavior of BN implies that the domain of convergence of the infinite product representation of the Dirichlet L-functions for non-principal characters can be extended from down to , without encountering any zeros before reaching this critical line.

On short products of primes in arithmetic progressions

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha, \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots, p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at most $\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).

On the regularity of primes in arithmetic progressions

We prove that for a positive integer [Formula: see text] the primes in certain kinds of intervals cannot distribute too “uniformly” among the reduced residue classes modulo [Formula: see text]. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.

Unexpected biases in the distribution of consecutive primes

Although the sequence of primes is very well distributed in the reduced residue classes [Formula: see text], the distribution of pairs of consecutive primes among the permissible ϕ(q)(2) pairs of reduced residue classes [Formula: see text] is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.

On generalizations of problems of Recaman and Pomerance

Answering a question of Balasubramanian, we find all primes p for which there exist p consecutive primes forming a complete residue system (mod p). On the other hand, under the prime ℓ-tuple conjecture we show that for any k≥2, there exist infinitely many sets of φ(k) consecutive primes forming reduced residue classes (mod k). The problems considered are generalizations of those of Recaman and Pomerance, respectively.

Prime number races with three or more competitors

Let \(q\ge 3\) and \(2\le r\le \phi (q)\) be positive integers, and \(a_1,\ldots ,a_r\) be distinct reduced residue classes modulo \(q\). Rubinstein and Sarnak defined \(\delta (q;a_1,\ldots ,a_r)\) to be the logarithmic density of the set of real numbers \(x\) such that \(\pi (x;q,a_1)>\pi (x;q,a_2)>\cdots >\pi (x;q,a_r)\). In this paper, we establish an asymptotic formula for \(\delta (q;a_1,\ldots ,a_r)\) when \(r\ge 3\) is fixed and \(q\) is large. Several applications concerning these prime number races are then deduced. First, comparing with a recent work of Fiorilli and Martin on the case \(r=2\), we show that these densities behave differently when \(r\ge 3\). Another surprising consequence of our results is that, unlike two-way races, biases do appear in races involving three or more squares (or non-squares) to large moduli. Furthermore, we establish a partial result towards a conjecture of Rubinstein and Sarnak on biased races, and disprove a recent conjecture of Feuerverger and Martin concerning bias factors. Lastly, we use our method to derive the Fiorilli and Martin asymptotic formula for the densities when \(r=2\).

On the least primitive root in number fields

Let K be an algebraic number field and \( \mathfrak{O} \)K its ring of integers. For any prime ideal \( \mathfrak{p} \), the group \( (\mathfrak{O}_K /\mathfrak{p})* \) of the reduced residue classes of integers is cyclic. We call any element of a generator of the group \( (\mathfrak{O}_K /\mathfrak{p})* \) a primitive root modulo \( \mathfrak{p} \). Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal \( \mathfrak{p} \) under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.

Applications for Cryptography of the Structure of the Group of Reduced Residue Classes of Residue Ring of Z/2^wZ(Application)

Applications for Cryptography of the Structure of the Group of Reduced Residue Classes of Residue Ring of Z/2^wZ(Application)

Davenport constant for finite abelian groups

For a finite abelian group G, we investigate the length of a sequence of elements of G that is guaranteed to have a subsequence with product identity of G. In particular, we obtain a bound on the length which takes into account the repetitions of elements of the sequence, the rank and the invariant factors of G. Consequently, we see that there are plenty of such sequences whose length could be much shorter than the best known upper bound for the Davenport constant of G, which is the least integer s such that any sequence of length s in G necessarily contains a subsequence with product identity. We also show that the Davenport constant for the multiplicative group of reduced residue classes modulo n is comparatively large with respect to the order of the group, which is φ(n),when n is in certain thin subsets of positive integers. This is done by studying the Carmichael’s lambda function, defined as the maximal multiplicative order of any reduced residue modulo n, along these subsets.

On a generalization of the Demǰanenko determinant

The Demǰanenko matrices are generalized with the averaged Bernoulli polynomials, and their determinants are computed by the Dedekind-Frobenius formula. This enables us to interpret geometrically the special values at negative integers of Dedekind zeta function of maximal real subfields as well as of imaginary subfields of cyclotomic fields. We utilize the structural properties of the group of reduced residue classes modm and those of (averaged) Bernoulli polynomials, thus appealing to the predominance of Galois theory over the fields themselves, which makes it possible to present very lucid reasoning of the whole picture.

The least primitive root in number fields

Let K be an algebraic number field and \( \mathfrak{O} \) K its ring of integers. For any prime ideal \( \mathfrak{p} \), the group \( (\mathfrak{O}_K /\mathfrak{p})* \) of the reduced residue classes of integers is cyclic. We call any element of a generator of the group \( (\mathfrak{O}_K /\mathfrak{p})* \) a primitive root modulo \( \mathfrak{p} \). Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal \( \mathfrak{p} \) under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.

Primes in Short Segments of Arithmetic Progressions

AbstractConsider the variance for the number of primes that are both in the interval [y,y + h] for y ∈ [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h/q ≤ x1/2-∈ , for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all” q in the range 1 ≤ h/q ≤ x1/4-∈, that on averaging over q one obtains an asymptotic formula in the extended range 1 ≤ h/q ≤ x1/2-∈, and that there are lower bounds with the correct order of magnitude for all q in the range 1 ≤ h/q ≤ x1/3-∈.

Uniform bounds for the least almost‐prime primitive root

We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the least prime or almost-prime primitive root (mod p) that holds uniformly for all p, analogous to Linnik's Theorem on a uniform upper bound for the least prime in a single arithmetic progression (mod p).

On a variance associated with the distribution of general sequences in arithmetic progressions. I

An asymptotic formula of Montgomery–Hooley type is established for general sequences which, for relatively small moduli, are approximately equidistributed in the reduced residue classes.

Greatest of the least primes in arithmetic progressions having a given modulus

We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about ϕ ( k ) log ⁡ k log ⁡ ϕ ( k ) \phi (k)\log k\log \phi (k) , where ϕ \phi is Euler’s phi-function.