Menon-type identities concerning Dirichlet characters

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.

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

Nagoya Mathematical Journal ◽

10.1017/s0027763000004906 ◽

1994 ◽

Vol 134 ◽

pp. 151-172 ◽

Cited By ~ 6

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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Non-random behavior in sums of modular symbols

International Journal of Number Theory ◽

10.1142/s1793042122500464 ◽

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.

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On primitivity of Dirichlet characters

International Journal of Number Theory ◽

10.1142/s1793042115500839 ◽

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.

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

Research in Number Theory ◽

10.1007/s40993-020-00213-1 ◽

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.

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Dirichlet characters of the rational polynomials

AIMS Mathematics ◽

10.3934/math.2022194 ◽

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>Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} </tex-math></disp-formula> 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.</abstract>

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A generalization of Menon’s identity with Dirichlet characters

International Journal of Number Theory ◽

10.1142/s1793042118501579 ◽

2018 ◽

Vol 14 (10) ◽

pp. 2631-2639 ◽

Cited By ~ 2

Author(s):

Yan Li ◽

Xiaoyu Hu ◽

Daeyeoul Kim

Keyword(s):

Number Theory ◽

Positive Integer ◽

Dirichlet Character ◽

Greatest Common Divisor ◽

Divisor Function ◽

Euler’S Totient Function ◽

Group Of Units ◽

Dirichlet Characters ◽

Indian Math ◽

Sum Formula

The classical Menon’s identity [P. K. Menon, On the sum [Formula: see text], J. Indian Math. Soc.[Formula: see text]N.S.[Formula: see text] 29 (1965) 155–163] states that [Formula: see text] where for a positive integer [Formula: see text], [Formula: see text] is the group of units of the ring [Formula: see text], [Formula: see text] represents the greatest common divisor, [Formula: see text] is the Euler’s totient function and [Formula: see text] is the divisor function. In this paper, we generalize Menon’s identity with Dirichlet characters in the following way: [Formula: see text] where [Formula: see text] is a non-negative integer and [Formula: see text] is a Dirichlet character modulo [Formula: see text] whose conductor is [Formula: see text]. Our result can be viewed as an extension of Zhao and Cao’s result [Another generalization of Menon’s identity, Int. J. Number Theory 13(9) (2017) 2373–2379] to [Formula: see text]. It can also be viewed as an extension of Sury’s result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo 58 (2009) 99–108] to Dirichlet characters.

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Ramanujan’s convolution sum twisted by Dirichlet characters

International Journal of Number Theory ◽

10.1142/s1793042119500027 ◽

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.

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