Character Modulo Research Articles

Second-order linear recurrences with identically distributed residues modulo p^e

Let $p$ be an odd prime and let $u(a,-1)$ and $u(a',-1)$ be two Lucas sequences whose discriminants have the same nonzero quadratic character modulo $p$ and whose periods modulo $p$ are equal. We prove that there is then an integer $c$ such that for all $d\in\mathbb Z_p$, the frequency with which $d$ appears in a full period of $u(a,-1)\pmod p$ is the same frequency as $cd$ appears in $u(a',-1)\pmod p$. Here $u(a,b)$ satisfies the recursion relation $u_{n+2}=au_{n+1}+bu_n$ with initial terms $u_0=0$ and $u_1=1$. Similar results are obtained for the companion Lucas sequences $v(a,-1)$ and $v(a',-1)$. This paper extends analogous statements for Lucas sequences of the form $u(a,1)\pmod p$ given in a previous article. We further generalize our results by showing for a certain class of primes $p$ that if $e>1$, $b=\pm 1$, and $u(a,b)$ and $u(a',b)$ are Lucas sequences with the same period modulo $p$, then there exists an integer $c$ such that for all residues $d\pmod{p^e}$, the frequency with which $d$ appears in $u(a,b)\pmod{p^e}$ is the same frequency as $cd$ appears in $u(a',b)\pmod{p^e}$.

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.

Identities of general Kloosterman sums

Let [Formula: see text] be any integers with [Formula: see text], and [Formula: see text] be a Dirichlet character modulo [Formula: see text]. The general Kloosterman sums [Formula: see text] are defined as follows: [Formula: see text] where [Formula: see text], and [Formula: see text] denotes the multiplicative inverse of [Formula: see text] modulo [Formula: see text]. By combining elementary methods and analytic methods, along with Montgomery and Vaughan’s clever trick on primitive characters, we derive some new identities for [Formula: see text] under the condition [Formula: see text], where [Formula: see text] denotes the summation over all [Formula: see text] with [Formula: see text]. Previously only the case of [Formula: see text] was studied.

A new Ramanujan-type identity for $$L(2k+1, \chi _1)$$

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan’s idea, we rediscovered a Ramanujan-type identity for \(\zeta (2k+1)\) that was first established by Malurkar and later by Berndt using different techniques. In the current paper, we extend the aforementioned identity of Malurkar and Berndt to derive a new Ramanujan-type identity for \(L(2k+1, \chi _1)\), where \(\chi _1\) is the principal character modulo prime p. In the process, we encounter a new family of Ramanujan-type polynomials. Furthermore, we establish a character analogue of Grosswald’s identity.

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

Let \(k \ge 2\) and N be positive integers and let \(\chi \) be a Dirichlet character modulo N. Let f(z) be a modular form in \(M_k(\Gamma _0(N),\chi )\). Then we have a unique decomposition \(f(z)=E_f(z)+S_f(z)\), where \(E_f(z) \in E_k(\Gamma _0(N),\chi )\) and \(S_f(z) \in S_k(\Gamma _0(N),\chi )\). In this paper, we give an explicit formula for \(E_f(z)\) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel’s formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities.

The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums

Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

Value-distribution of quartic Hecke L-functions

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $\sigma>1/2$, we prove an asymptotic distribution function $F_{\sigma}$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,\chi_c)L(s,\overline{\chi}_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_{\sigma}$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$.

Small Values of |L'/L(1,χ)|

In this article, we investigate the quantity , as over the primes, where is the Dirichlet L-function attached to a non-principal Dirichlet character modulo q. Our main result shows that . We also compute mq for every odd prime q up to 10. As a consequence we numerically verified that for every prime q, , we have , with . In particular, this shows that for every non-principal Dirichlet character χ mod q where is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the mq-values. The programs used and the computational results described here are available at the following web address: https://www.math.unipd.it/~languasc/smallvalues.html.

An asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$

Let $q$ be a positive integer ($\geq 2$), $\chi$ be a Dirichlet character modulo $q$, $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^{\prime} (s, \chi)$ denote its derivative with respect to the complex variable $s$. Let $t_{0}$ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_{0} = 0$).

An investigation into explicit versions of Burgess' bound

Let χ be a Dirichlet character modulo p, a prime. In applications, one often needs estimates for short sums involving χ. One such estimate is the family of bounds known as Burgess' bound. In this paper, we explore several adjustments one can make to the work of Treviño [11] in obtaining explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a kth power non-residue modulo p which is less than pα for several fixed α. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7].

Exceptional characters and nonvanishing of Dirichlet L-functions

Let \(\psi \) be a real primitive character modulo D. If the L-function \(L(s,\psi )\) has a real zero close to \(s=1\), known as a Landau–Siegel zero, then we say the character \(\psi \) is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values \(L(1/2,\chi )\) of the Dirichlet L-functions \(L(s,\chi )\) are nonzero, where \(\chi \) ranges over primitive characters modulo q and q is a large prime of size \(D^{O(1)}\). Under the same hypothesis we also show that, for almost all \(\chi \), the function \(L(s,\chi )\) has at most a simple zero at \(s = 1/2\).

Complex numbers similar to the generalized Bernoulli numbers and their applications

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

Some identities involving Gauss sums

<abstract><p>We calculate several identities involving some Gauss sums of the $ 2^k $-order character modulo an odd prime $ p $ by using the elementary and analytic methods, and finally give several exact and interesting formulae for them. The properties of the classical Gauss sums play an important role in the proof of this paper.</p></abstract>

A Pólya–Vinogradov inequality for short character sums

AbstractIn this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$ with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.

Quantitative non-vanishing of Dirichlet L-values modulo p

Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that $$p \not \mid N$$ and $$\lambda $$ a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character $$\chi $$ modulo F with the critical Dirichlet L-value $$L(-k,\lambda \chi )$$ being p-indivisible. Here $$F \rightarrow \infty $$ with $$(N,F)=1$$ and $$p\not \mid F\phi (F)$$ . We explore the indivisibility via an algebraic and a homological approach. The latter leads to a bound of the form $$F^{1/2}$$ . The p-indivisibility yields results on the distribution of the associated p-Selmer ranks. We also consider an Iwasawa variant. It leads to an explicit upper bound on the lowest conductor of the characters factoring through the Iwasawa $${{\mathbb {Z}}}_{\ell }$$ -extension of $${{\mathbb {Q}}}$$ for an odd prime $$\ell \ne p$$ with the corresponding critical L-value twists being p-indivisible.

Mean-square estimate of automorphic L-functions

Let f be a holomorphic Hecke cusp form with even integral weight k ⩾ 2 for the full modular group, and let χ be a primitive Dirichlet character modulo q. Let Lf (s, χ) be the automorphic L-function attached to f and χ. We study the mean-square estimate of Lf (s, χ) and establish an asymptotic formula.

Обобщенная формула следа Петерсона для конгруэнцподгупп

In this paper, a generalization of the Petersson trace formula for holomorphic parabolic forms with positive even weight with respect to congruence subgroup \Gamma_0 (N) of the Dirichlet character modulo N is proved

An elementary bound on Siegel zeroes

We consider Dirichlet L-functions L(s,χ) where χ is a real, non-principal character modulo q. Using Pintz's refinement of Page's theorem, we prove that for q≥3 the function L(s,χ) has at most one real zero β with 1−0.933/log⁡q<β<1.

On asymptotic properties of the generalized Dirichlet L-functions

Let [Formula: see text] be an integer, [Formula: see text] denote a Dirichlet character modulo [Formula: see text], for any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-functions [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] by analytic continuation. In this paper, we study the mean value properties of the generalized Dirichlet [Formula: see text]-functions, and obtain several sharp asymptotic formulae by using analytic method.

Hybrid subconvexity bounds for twisted L-functions on GL(3)

Let $q$ be a large prime, and $\chi$ the quadratic character modulo $q$. Let $\phi$ be a self-dual Hecke--Maass cusp form for $SL(3,\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $\Gamma_0(q)\subseteq SL(2,\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \[ L(1/2,\phi\times u_j\times\chi)\ll_{\phi,\varepsilon} (qt_j)^{3/2-\theta+\varepsilon},\quad L(1/2+it,\phi\times\chi)\ll_{\phi,\varepsilon} (qt)^{3/4-\theta/2+\varepsilon}, \] for any $\varepsilon>0$, where $\theta=1/23$ is admissible.