On the Sizes of Gaps in the Fourier Expansion of Modular Forms

2005 ◽  
Vol 57 (3) ◽  
pp. 449-470 ◽  
Author(s):  
Emre Alkan
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
The Riemann Zeta Function ◽  
Classical Paper ◽  
Riemann Zeta ◽  
Image Position ◽  
Almost All ◽  
Integer Weight

AbstractLet be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letConcerning bounded values of i f (n) we prove that for ∊ > 0 there exists M = M(∊, f ) such that Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then . Using a result of David and Pappalardi, we improve the exponent to for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate i f (n) on the average using well known bounds on the Riemann Zeta function.

On the Riemann zeta-function

1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Absolute Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.

AVERAGE SIZE OF GAPS IN THE FOURIER EXPANSION OF MODULAR FORMS

2007 ◽  
Vol 03 (02) ◽  
pp. 207-215 ◽  
Author(s):  
EMRE ALKAN
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
Gap Function ◽  
Quantitative Results ◽  
Average Size

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function.

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

Lambert series associated to Hilbert modular form

2021 ◽  
pp. 1-15
Author(s):  
Rishabh Agnihotri
Keyword(s):  
Asymptotic Expansion ◽  
Modular Form ◽  
Cusp Form ◽  
Riemann Zeta Function ◽  
Hilbert Modular Form ◽  
Lambert Series ◽  
Nontrivial Zeros ◽  
Hilbert Modular ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

scholarly journals THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

2020 ◽  
Vol 8 ◽  
Author(s):  
BRAD RODGERS ◽  
TERENCE TAO
Keyword(s):  
Pair Correlation ◽  
Local Equilibrium ◽  
Local Distribution ◽  
Distribution Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
De Bruijn ◽  
Image Position ◽  
Argument Proceeds ◽  
Strong Control

For each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$ . The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$ , and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $\unicode[STIX]{x1D6EC}<t\leqslant 0$ , until one establishes that the zeros of $H_{0}$ are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

scholarly journals Distribution of unitarily k-free integers

1975 ◽  
Vol 20 (2) ◽  
pp. 129-141 ◽  
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Error Term ◽  
Real Variable ◽  
Canonical Representation ◽  
Order Estimate ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
The Riemann Zeta Function ◽  
Elementary Methods ◽  
Riemann Zeta ◽  
Image Position

Let k be a fixed integer ≧2. A positive integer n is called unitarily k-free, if the multiplicity of each prime divisor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the kth power of any integer > 1. By a unitary divisor, we mean as usual a divisor d > 0 of n such that (d,(n/d)) = 1. The integer 1 is also considered to be unitarily k-free. These integers were first defined by Cohen (1961; § 1). Let Q*k denote the set of unitarily k-free integers. When k = 2, the set Q*2 coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1961; § 1 and § 6). Let x denote a real variable 1 and let Q*k denote the number of unitarily k-free integers ≦ x. Cohen (1961; Theorem 3.2) established by purely elementary methods that , where , the product being extended over all primes p and ζ(k) denotes the Riemann Zeta function. In the same paper Cohen (1961; Theorem 4.2) improved the order estimate of the error term in (1.1) to O(x1/k), by making use of the properties of real Dirichiet series. Later, he (Cohen; 1964) proved the same result by purely elementary methods eliminating the use of Dirichlet series.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

A simple series representation for Apéry's constant

2013 ◽  
Vol 97 (540) ◽  
pp. 455-460 ◽  
Author(s):  
John Melville
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Quantum Electrodynamics ◽  
Series Representation ◽  
Gyromagnetic Ratio ◽  
French Mathematician ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. ThusThis constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

scholarly journals Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

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