Character Modulo Research Articles

On the Dirichlet characters of polynomials in several variables

x=N+1 χ(f(x)) ≪ p log p, where A ≪ B denotes |A| < kB for some constant k which in this case depends on the degree of f . The estimate (1) is the best possible. In fact, the first author [8] proved the following: Let q ≥ 3 be an integer, χ a primitive character modulo q, and m and n two positive integers such that χm, χn and χm+n are also primitive characters modulo q. Then for any integers r and s with (r − s, q) = 1, we have the identity

On the mean value of [formula omitted

Let χ be the Dirichlet character modulo q ⩾ 3 and L ( s , χ ) denote the corresponding Dirichlet L-function. The mean value of | L ′ L ( 1 , χ ) | 2 k is studied and a few asymptotic formulae are given. Hybrid mean value of L ′ L ( 1 , χ ) , general Kloosterman sums and general quadratic Gauss sums are considered.

A CLASSICAL CHARACTERIZATION OF NEWFORMS WITH EQUIVALENT EIGENFORMS IN ${S_{k + 1/2}}(4N, \chi)$

The paper continues the investigation of the Hecke structure of spaces of half-integral weight cusp forms ${S_{k + 1/2}}(4N, \chi)$ , where $k$ and $N$ are positive integers with $N$ odd, and $\chi$ is an even quadratic Dirichlet character modulo $4N$ . In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non-twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in ${S_{k + 1/2}}(4N, \chi)$ . The result reformulates, in classical terms, the representation-theoretic conditions given by Flicker and Waldspurger. The conditions involve easily-verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half-integral weight cusp forms in any space ${S_{k + 1/2}}(4N, \chi)$ as above.

Infinitely Divisible Laws Associated with Hyperbolic Functions

AbstractThe infinitely divisible distributions on of random variables Ct, St and Tt with Laplace transformsrespectively are characterized for various t &gt; 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of C1 and S2 are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet L-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from St and Ct by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

On Exponential Sums Involving the Ideal Counting Function in Quadratic Number Fields

Let χ be a Dirichlet character modulo k > 1, and Fχ(n) the arithmetical function which is generated by the product of the Riemann zeta-function and the Dirichlet L-function corresponding to χ in \(\). In this paper we study the asymptotic behaviour of the exponential sums involving the arithmetical function Fχ(n). In particular, we study summation formulas for these exponential sums and mean square formulas for the error term.

Twisted higher moments of Kloosterman sums

Let X be a nontrivial Dirichlet character modulo an odd prime p. Write S(a) = e( = 2 p cos θ(α). We shall prove formula math. and, for complex X, formula math where c(k) is a constant depending only on k.

Hecke theory of series formed with modular symbols and relations among convolution L -functions

In [G1], [G2], the distribution of modular symbols is studied and a new class of functions which satisfy a transformation law involving these objects is introduced. The goal of Goldfeld’s program is to prove Szpiro’s conjecture which states that for elliptic curves with minimal discriminant D and conductor N there is an absolute constant κ such that D ? N. To do this, an equivalent conjecture involving modular symbols is established in [G4] using the, now proven, conjecture of Shimura, Taniyama and Weil. It seems that a sufficiently good understanding of the new series proposed by Goldfeld should yield a resolution of these conjectures. We repeat here their definition in a somewhat more general form to include period polynomials rather than modular symbols only. For positive integers M,N such that M |N, let m, k be non-negative integers such that m ≥ k − 2 ≥ 0 and let χ be a Dirichlet character modulo N. First, for each f ∈ Sk(M) = {holomorphic cusp forms of weight k and level M} we denote by rf the map which sends γ ∈ Γ0(M), the Hecke congruence group of level M , to the polynomial function:

Hecke structure of spaces of half-integral weight cusp forms

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp formsSk/2(4N,χ) wherekandNare odd nonnegative integers withk≥ 3, andχis an even quadratic Dirichlet character modulo 4N. We give complete results in the case whereNis a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions onNandχ, by giving conditions under which a newform has an equivalent cusp form inSk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any suchNandχ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Zeroes of Dirichlet $L$-functions and irregularities in the distribution of primes

Seven widely spaced regions of integers with π4,3(x) < π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of π4,3(x) - π4,1(x) including the sign change (axis crossing) regions, in time linear in x, using zeroes of L(s,x),x the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to 10 1000 Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference li(x)-π(x) with fine resolution out to and beyond the region in the vicinity of 6.658 x 10 370 discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of 1.39822 × 10 316 .

On the theory of newforms of half-integral weight

In this paper we will study the theory of newforms in S k + 1 2 (Γ 0(4M),χ 1) , for M an odd squarefree natural number, χ 1 = ( 4ε · )χ , where χ is a quadratic character modulo M with χ(−1) = ε, parallel to the Atkin-Lehner theory of newforms in S 2 k (2 M) and prove that these two spaces are isomorphic under certain linear combinations of Shimura liftings, commuting with the action of Hecke operators.

The mean square of the Dedekind zeta function in quadratic number fields

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually proveThis may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

The Zeroes of Functions Related to Dirichlet L-Functions

Hecke, [3], has shown for x a real Dirichlet character modulo q, the associated Dirichlet L-function L(s, x) has infinitely many zeroes on the line Here, using a method of Polya, [5], we show that both the real and imaginary parts of a function associated to L(s, x) through the functional equation, have infinitely many zeroes on any line

Some numerical results on Fekete polynomials

It is known that if χ \chi is a real residue character modulo k with χ ( p ) = − 1 \chi (p) = - 1 for the first five primes p, then the corresponding Fekete polynomial Σ n = 1 k χ ( n ) x n \Sigma _{n = 1}^k\;\chi (n){x^n} changes sign on (0, 1). In this paper it is shown that the condition that χ ( p ) \chi (p) be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if χ \chi is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though χ ( 2 ) = χ ( 3 ) = χ ( 5 ) = χ ( 7 ) = − 1 \chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1 .

Some Numerical Results on Fekete Polynomials

It is known that if $\chi$ is a real residue character modulo k with $\chi (p) = - 1$ for the first five primes p, then the corresponding Fekete polynomial $\Sigma _{n = 1}^k\;\chi (n){x^n}$ changes sign on (0, 1). In this paper it is shown that the condition that $\chi (p)$ be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if $\chi$ is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though $\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1$.

Examples of modular forms

The Fourier coefficients of certain automorphic forms are evaluated using a tensor product construction appearing in a formal proof of the Ramanujan hypothesis [ 11. Let r be the set of two-by-two matrices with integer entries and positive determinant. The modular group is the set of elements of I’ of determinant one. A subgroup of the modular group is the set r( 1) of elements having two even entries. For each positive integer r, let I’(r) be the subgroup of r(l) formed by the elements whose lower left entry is divisible by r. A character modulo r is a function x(n) of integral 1z which is periodic of period r, which satisfies the identity x(mn) = x(m) x(n) f or all integers m and n, which has nonzero values at points which are relatively prime to r, and which vanishes at points which are not relatively prime to r. A character is either an even or an odd function. The signature for r(l) is the homomorphism of r(l) onto the fourth roots of unity which is 1 on matrices which are congruent modulo four to

An elementary proof of some character sum identities of Apostol

Let х denote a primitive character modulo k. Using two different representations for Dirichlet L-functions, Apostol [1] recently derived a representation forinvolving the sumswhere m is a positive integer. Furthermore, if х(r) = (r|p)the residue class character modulo the odd prime p, he derived a representation for Mm(х) involving the sums

Remarks concerning the values of the riemann zeta function at integral, odd arguments

It is known that for a = 2 n + 1, ζ( a) = π a r( a) − G a ( i), where G a ( τ) has a Fourier expansion G a ( τ) = Σ n = 1 ∞ c n ( a) e 2 πinτ and r( a) is rational. For a slightly more general function G a(τ, χ) = Σ n = 1 ∞ c n(a, χ) e 2πinτ k [which, for k = 1 reduces to G a ( τ)] it is shown that if χ( n) is a nonprincipal, primitive congruence character modulo k, then G a(i, χ) = π aτ( χ)r(a, χ) . Here χ is the conjugate character, τ( χ) is a Gaussian sum, and r( a, χ) is algebraic. If χ( n) is also real, then k −1 2 π −aG a(i, χ) is actually rational.