Non-random behavior in sums of modular symbols

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

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.

A note on prime number races and zero free regions for L functions

Let [Formula: see text] be a real and non-principal Dirichlet character, [Formula: see text] its Dirichlet [Formula: see text]-function and let [Formula: see text] be a generic prime number. We prove the following result: If for some [Formula: see text] the partial sums [Formula: see text] change sign only for a finite number of integers [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] has no zeros in the half plane [Formula: see text].

A short note on inadmissible coefficients of weight 2 and $$2k+1$$ newforms

Let $$f(z)=q+\sum _{n\ge 2}a(n)q^n$$ f ( z ) = q + ∑ n ≥ 2 a ( n ) q n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for $$k=2$$ k = 2 by ruling out or locating all odd prime values $$|\ell |<100$$ | ℓ | < 100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights $$k\ge 1$$ k ≥ 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

Symmetric identity for polynomial sequences satisfying A′ n +1(x) = (n + 1)An (x)

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

Central values of additive twists of cuspidal L-functions

Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

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.

Dirichlet characters of the rational polynomials

<abstract><p>Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} </tex-math></disp-formula></p> <p>where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity ^{[<xref ref-type="bibr" rid="b6">6</xref>]} under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.</p></abstract>

A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

<p style='text-indent:20px;'>Using elementary methods, we count the quadratic residues of a prime number of the form <inline-formula><tex-math id="M2">\begin{document}$ p = 4n-1 $\end{document}</tex-math></inline-formula> in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> of the imaginary quadratic field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document}</tex-math></inline-formula> Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.</p>