Ramanujan’s convolution sum twisted by Dirichlet characters

2019 ◽  
Vol 15 (01) ◽  
pp. 137-152
Author(s):  
Zafer Selcuk Aygin ◽  
Nankun Hong
Keyword(s):  
Modular Forms ◽  
Dirichlet Character ◽  
Character Formula ◽  
Divisor Functions ◽  
Dirichlet Characters ◽  
Ramanujan’S Formula

We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character [Formula: see text], which are analogous to Ramanujan’s formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results.

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 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.

Ramanujan’s Formula and Modular Forms

2002 ◽  
pp. 159-212 ◽  
Author(s):  
Shigeru Kanemitsu ◽  
Yoshio Tanigawa ◽  
Masami Yoshimoto
Keyword(s):  
Modular Forms ◽  
Ramanujan’S Formula

scholarly journals Stark–Heegner points and the Shimura correspondence

2008 ◽  
Vol 144 (5) ◽  
pp. 1155-1175 ◽  
Author(s):  
Henri Darmon ◽  
Gonzalo Tornaría
Keyword(s):  
Modular Forms ◽  
Dirichlet Character ◽  
Heegner Points ◽  
Real Quadratic Fields ◽  
Half Integral Weight ◽  
Hida Family ◽  
Integral Weight ◽  
Similar Kind ◽  
Shimura Correspondence

AbstractLet $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

scholarly journals Menon-type identities concerning Dirichlet characters

2018 ◽  
Vol 14 (04) ◽  
pp. 1047-1054 ◽  
Author(s):  
László Tóth
Keyword(s):  
Dirichlet Character ◽  
Principal Character ◽  
Dirichlet Characters ◽  
Alternative Approach ◽  
Ramanujan Sums

Let [Formula: see text] be a Dirichlet character (mod [Formula: see text]) with conductor [Formula: see text]. In a quite recent paper Zhao and Cao deduced the identity [Formula: see text], which reduces to Menon’s identity if [Formula: see text] is the principal character (mod [Formula: see text]). We generalize the above identity by considering even functions (mod [Formula: see text]), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums.

scholarly journals ON SHIMURA'S DECOMPOSITION

2013 ◽  
Vol 09 (06) ◽  
pp. 1431-1445 ◽  
Author(s):  
SOMA PURKAIT
Keyword(s):  
Modular Forms ◽  
Theta Series ◽  
Dirichlet Character ◽  
Practical Applications ◽  
Quadratic Twists ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Sum Formula ◽  
Practical Algorithm ◽  
Definition Of

Let k be an odd integer ≥ 3 and N be a positive integer such that 4|N. Let χ be an even Dirichlet character modulo N. Shimura decomposes the space of half-integral weight cusp forms Sk/2(N,χ) as a direct sum [Formula: see text] where F runs through all newforms of weight k - 1, level dividing N/2 and character χ2, the space Sk/2(N,χ,F) is the subspace of forms that are "Shimura equivalent" to F, and the space S0(N,χ) is the subspace spanned by single-variable theta-series. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating the critical values of L-functions of quadratic twists of newforms of even integral weight to coefficients of modular forms of half-integral weight. In this paper, we give a more precise definition of the summands Sk/2(N,χ,F) whilst proving that it is equivalent to Shimura's definition. We use our definition to give a practical algorithm for computing Shimura's decomposition, and illustrate this with some examples.

On primitivity of Dirichlet characters

2015 ◽  
Vol 11 (06) ◽  
pp. 1913-1939
Author(s):  
R. Daileda ◽  
N. Jones
Keyword(s):  
Functional Equation ◽  
Dirichlet Character ◽  
Gauss Sum ◽  
Primitive Character ◽  
Dirichlet Characters

Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive. In this paper, we introduce a modification of "inducing to higher level" which causes imprimitive characters to behave primitively, in the sense that the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character.

scholarly journals Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Alessandro Languasco
Keyword(s):  
Dirichlet Character ◽  
Arithmetic Progressions ◽  
Cyclotomic Fields ◽  
Efficient Computation ◽  
Root Of Unity ◽  
Dirichlet Characters ◽  
The Difference

AbstractWe introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants $${\mathfrak {G}}_q$$ G q for the prime cyclotomic fields $$ {\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , where q is an odd prime and $$\zeta _q$$ ζ q is a primitive q-root of unity. With such a new algorithm we evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}_q^+$$ G q + , where $${\mathfrak {G}}_q^+$$ G q + is the Euler–Kronecker constant of the maximal real subfield of $${\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , for some very large primes q thus obtaining two new negative values of $${\mathfrak {G}}_q$$ G q : $${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$ G 9109334831 = - 0.248739 ⋯ and $${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$ G 9854964401 = - 0.096465 ⋯ We also evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + for every odd prime $$q\le 10^6$$ q ≤ 10 6 , thus enlarging the size of the previously known range for $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + . Our method also reveals that the difference $${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$ G q - G q + can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed $$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$ M q = max χ ≠ χ 0 | L ′ / L ( 1 , χ ) | for every odd prime $$q\le 10^6$$ q ≤ 10 6 , where $$L(s,\chi )$$ L ( s , χ ) are the Dirichlet L-functions, $$\chi $$ χ run over the non trivial Dirichlet characters mod q and $$\chi _0$$ χ 0 is the trivial Dirichlet character mod q. As another by-product of our computations, we will provide more data on the generalised Euler constants in arithmetic progressions.

scholarly journals Dirichlet characters of the rational polynomials

2021 ◽  
Vol 7 (3) ◽  
pp. 3494-3508
Author(s):  
Wenjia Guo ◽  
◽  
Xiaoge Liu ◽  
Tianping Zhang
Keyword(s):  
Dirichlet Character ◽  
Character Sums ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Dirichlet Characters ◽  
Fourth Power Mean ◽  
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 <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> 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>

scholarly journals Slopes of the U7 operator acting on a space of overconvergent modular forms

2012 ◽  
Vol 15 ◽  
pp. 113-139 ◽  
Author(s):  
L. J. P. Kilford ◽  
Ken McMurdy
Keyword(s):  
Modular Forms ◽  
Dirichlet Character ◽  
Root Of Unity ◽  
Primitive Dirichlet Character

AbstractLet χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

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