Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

Non-random behavior in sums of modular symbols

2021 ◽  
pp. 1-25
Author(s):  
Alex Cowan
Keyword(s):  
Eisenstein Series ◽  
Spectral Decomposition ◽  
Fourier Coefficients ◽  
Dirichlet Character ◽  
Common Phenomenon ◽  
Modular Symbols ◽  
Dirichlet Characters ◽  
Random Behavior ◽  
The Common ◽  
Automorphic Functions

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.

scholarly journals Dedekind sums arising from newform Eisenstein series

2020 ◽  
Vol 16 (10) ◽  
pp. 2129-2139
Author(s):  
T. Stucker ◽  
A. Vennos ◽  
M. P. Young
Keyword(s):  
Eisenstein Series ◽  
Short Proof ◽  
Explicit Construction ◽  
Transformation Rule ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Dirichlet Characters

For primitive nontrivial Dirichlet characters [Formula: see text] and [Formula: see text], we study the weight zero newform Eisenstein series [Formula: see text] at [Formula: see text]. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of [Formula: see text]. We also give a short proof of the reciprocity formula for this Dedekind sum.

scholarly journals The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields

1994 ◽  
Vol 134 ◽  
pp. 151-172 ◽  
Author(s):  
Masanori Katsurada ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Dirichlet Character ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Mean Values ◽  
Integer Points ◽  
Principal Character ◽  
Dirichlet Characters ◽  
The Mean

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We putwhere χ runs over all Dirichlet characters mod q except for the principal character χ0.

scholarly journals Dirichlet Characters, Gauss Sums, and Inverse Z Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Jing Gao ◽  
Huaning Liu
Keyword(s):  
Error Estimate ◽  
Unit Circle ◽  
Series Representation ◽  
Gauss Sums ◽  
General Algorithm ◽  
Möbius Transform ◽  
Dirichlet Characters

A generalized Möbius transform is presented. It is based on Dirichlet characters. A general algorithm is developed to compute the inverseZtransform on the unit circle, and an error estimate is given for the truncated series representation.

Another generalization of Menon’s identity

2017 ◽  
Vol 13 (09) ◽  
pp. 2373-2379 ◽  
Author(s):  
Xiao-Peng Zhao ◽  
Zhen-Fu Cao
Keyword(s):  
Euler’S Totient Function ◽  
Number Of Divisors ◽  
Dirichlet Characters

Let [Formula: see text] be the Euler’s totient function and [Formula: see text] be the number of divisors of [Formula: see text]. Menon’s beautiful identity states that [Formula: see text] Here we extend this identity to Dirichlet characters mod [Formula: see text].

scholarly journals CHARACTER AND OBJECT

2016 ◽  
Vol 9 (3) ◽  
pp. 480-510 ◽  
Author(s):  
JEREMY AVIGAD ◽  
REBECCA MORRIS
Keyword(s):  
Nineteenth Century ◽  
Arithmetic Progression ◽  
Common Factor ◽  
Higher Order ◽  
Gradual Transition ◽  
Original Proof ◽  
Ordinary Objects ◽  
Mode Of Presentation ◽  
Dirichlet Characters ◽  
History Of

AbstractIn 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing overDirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment.

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