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 Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

2020 ◽  
Vol 7 (1) ◽  
pp. 81-101
Author(s):  
Issa Zabsonre ◽  
Djendode Mbainadji
Keyword(s):  
Phase Space ◽  
Spectral Decomposition ◽  
Measure Theory ◽  
New Approach ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Automorphic ◽  
Automorphic Functions

AbstractUsing the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.

scholarly journals A distinct negative leader propagation mode

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
O. Scholten ◽  
B. M. Hare ◽  
J. Dwyer ◽  
N. Liu ◽  
C. Sterpka ◽  
...  
Keyword(s):  
Radio Telescope ◽  
Electric Discharge ◽  
Radio Astronomy ◽  
Propagation Mode ◽  
Common Phenomenon ◽  
High Charge ◽  
Initial Stage ◽  
Lightning Initiation ◽  
The Common

AbstractThe common phenomenon of lightning still harbors many secrets such as what are the conditions for lightning initiation and what is driving the discharge to propagate over several tens of kilometers through the atmosphere forming conducting ionized channels called leaders. Since lightning is an electric discharge phenomenon, there are positively and negatively charged leaders. In this work we report on measurements made with the LOFAR radio telescope, an instrument primarily build for radio-astronomy observations. It is observed that a negative leader rather suddenly changes, for a few milliseconds, into a mode where it radiates 100 times more VHF power than typical negative leaders after which it spawns a large number of more typical negative leaders. This mode occurs during the initial stage, soon after initiation, of all lightning flashes we have mapped (about 25). For some flashes this mode occurs also well after initiation and we show one case where it is triggered twice, some 100 ms apart. We postulate that this is indicative of a small (order of 5 km$$^2$$ 2 ) high charge pocket. Lightning thus appears to be initiated exclusively in the vicinity of such a small but dense charge pocket.

scholarly journals Eisenstein series twisted by modular symbols for the group $\SL _{n}$

2000 ◽  
Vol 7 (6) ◽  
pp. 747-756 ◽  
Author(s):  
Dorian Goldfeld ◽  
Paul E. Gunnells
Keyword(s):  
Eisenstein Series ◽  
Modular Symbols

scholarly journals Kloosterman sums and Fourier coefficients of Eisenstein series

2018 ◽  
Vol 49 (2) ◽  
pp. 391-409 ◽  
Author(s):  
Eren Mehmet Kıral ◽  
Matthew P. Young
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Kloosterman Sums

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.

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