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On the zeros of period functions associated to the Eisenstein series for \(\Gamma_0^+(N)\). (English) Zbl 1491.11043

For a positive integer \(N\), let \(\Gamma_0(N)^+\) be the modular group generated by the elements of \(\Gamma_0(N)\) and the Fricke involution \(W_N\). Let \(E_{k}^+(z)\) be the Eisenstein series of even weight \(k\) for \(\Gamma_0(N)^+\). For a modular form \(f(z)=\sum_{n=0}^\infty a(n)e^{2\pi inz}\) of weight \(k\), the Eichler integral of \(f\) is defined by \(\mathcal{E}_f(z)=\sum_{n=0}^\infty a(n)n^{1-k}e^{2\pi inz}\). As an analogy of the period function of Eisenstein series for \(\mathrm{SL}_2(\mathbb Z)\), the authors define the period function of \(E_{k}^+\) by \(r(E_{k}^+;z)=-\frac{\Gamma(k-1)}{(2\pi i)^{k-1}}(\mathcal{E}_{E_{k}^+}-\mathcal{E}_{E_{k}^+}|_{2-k}W_N(z))\). The function \(r(E_{k}^+;z)-((k-1)N^{k/2}z)^{-1}\) is a polynomial of \(z\) of degree \(k-1\). From \(r(E_{k}^+;z)\), they consider three polynomials \(R_k(z)=(k-1)zr(E_{k}^+;z), R_k^-(z)=(k-1)zr^-(E_{k}^+;z)\) and \(P_k(z)\), where \(r^-(E_{k}^+;z)\) is the odd part of \(r(E_{k}^+;z)\) and, up to a nonzero constant, \(P_k(z)\) is defined by \(r(E_{k}^+;z)-(z^{k-1}+\frac{1}{N^{k/2}z})/(k-1)\). The polynomials \(R_k^-\) and \(R_k\) are of degree \(k\) and \(P_k\) is of degree \(k-2\).
The authors are interested in the location of zeros of these three polynomials and show that all zeros of \(P_k\) (resp. \(R_k^-\) and \(R_k\)) lie on the circle \(|z|=1/\sqrt{N}\), if \(N\ge 3, k\ge 4\) (resp. \(N\ge 5, k\ge 10 \) and \(N\ge 17, k\ge 10\)), and the zeros of each polynomial are simple and regularly distributed on this circle. Further, if \(N=1,k\ge 4\) and \(N=2, k\ge 1\), then all zeros of \(P_k\) lie on the circle \(|z|=1/\sqrt{N}\). If \(N=2,3,4\) and \(k\ge 4\), then \(R_k^-\) has exactly \(k-4\) zeros lying on the circle \(|z|=1/\sqrt N\) and four distinct real zeros lying outside the circle. The key point of the proof is the \(N\)-self-inversive property of \(P_k,R_k\) and \(R_k^-\). Here a polynomial \(P(z)\) of degree \(d\) is \(N\)-self-inversive if \(P\) satisfies \( P(z)=\epsilon (\sqrt Nz)^dP(1/Nz)\) for some \(\epsilon\in\mathbb C\). For \(N\)-self-inversive polynomial \(P(z)=\sum_{j=0}^dA_jz^j\in\mathbb C[z]\), if \(|A_d|\ge \frac12 \sum_{j=1}^{d-1}|A_j|\sqrt N^{d-1}\), then all of zeros of \(P\) lie on the circle \(|z|=1/\sqrt N\). Therefore, they check the above inequality for the coefficients of the polynomials under consideration. Further to show that the zeros are simple and regularly distributed on the circle, they deal with many fine inequalities.

MSC:

11F03 Modular and automorphic functions
11F11 Holomorphic modular forms of integral weight
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References:

[1] Berndt, Bruce C.; Straub, Armin, Ramanujan’s formula for \(\zeta(2 n + 1)\), (Exploring the Riemann Zeta Function (2017), Springer), 13-34 · Zbl 1497.11198
[2] Bringmann, Kathrin; Guerzhoy, Pavel; Kent, Zachary; Ono, Ken, Eichler-Shimura theory for mock modular forms, Math. Ann., 355, 3, 1085-1121 (2013) · Zbl 1312.11039
[3] Choi, SoYoung; Kim, Chang Heon, Mock modular period functions and L-functions of cusp forms in higher level cases, Proc. Am. Math. Soc., 142, 10, 3369-3386 (2014) · Zbl 1305.11029
[4] Choi, S.; Kim, C. H., Mock period functions in higher level cases, J. Math. Anal. Appl., 415, 499-512 (2014) · Zbl 1342.11056
[5] Conrey, John Brian; Farmer, David W.; Imamoglu, Özlem, The nontrivial zeros of period polynomials of modular forms Lie on the unit circle, Int. Math. Res. Not. IMRN, 2013, 20, 4758-4771 (2013) · Zbl 1305.11030
[6] Diamantis, Nikolaos; Rolen, Larry, Period polynomials, derivatives of L-functions, and zeros of polynomials, Res. Math. Sci., 5, 1, Article 9 pp. (2018) · Zbl 1457.11032
[7] El-Guindy, Ahmad; Raji, Wissam, Unimodularity of zeros of period polynomials of Hecke eigenforms, Bull. Lond. Math. Soc., 46, 3, 528-536 (2014), (English summary) · Zbl 1311.11027
[8] Jin, S.; Ma, W.; Ono, K.; Soundararajan, K., Riemann hypothesis for period polynomials of modular forms, Proc. Natl. Acad. Sci. USA, 113, 10, 2603-2608 (2016) · Zbl 1412.11075
[9] Kohnen, W.; Zagier, D., Modular forms with rational periods, (Rankin, R. A., Modular Forms (1984), Ellis Horwood), 197-249 · Zbl 0618.10019
[10] Lakatos, Piroska; Losonczi, László, Self-inversive polynomials whose zeros are on the unit circle, Publ. Math. (Debr.), 65, 3-4, 409-420 (2004) · Zbl 1150.30311
[11] Lalín, Matilde N.; Rogers, Mathew D., Variations of the Ramanujan polynomials and remarks on \(\zeta(2 j + 1) / \pi^{2 j + 1}\), Funct. Approx. Comment. Math., 48, 1, 91-111 (2013) · Zbl 1272.26007
[12] Lalín, Matilde N.; Smyth, Chris J., Unimodularity of zeros of self-inversive polynomials, Acta Math. Hung., 138, 1-2, 85-101 (2013) · Zbl 1299.26037
[13] Lalín, Matilde N.; Smyth, Chris J., Addendum to: unimodularity of zeros of self-inversive polynomials, Acta Math. Hung., 147, 1, 255-257 (2015) · Zbl 1363.26022
[14] Ram Murty, M.; Smyth, Chris; Wang, Rob J., Zeros of Ramanujan polynomials, J. Ramanujan Math. Soc., 26, 1, 107-125 (2011) · Zbl 1238.11033
[15] Nordentoft, Asbjørn Christian, On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials, Int. Math. Res. Not. (2020), rnaa 194 · Zbl 1477.11094
[16] Schinzel, A., Self-inversive polynomials with all zeros on the unit circle, Ramanujan J., 9, 1, 19-23 (2005) · Zbl 1079.30004
[17] Zagier, D., Periods of modular forms, traces of Hecke operators and multiple zeta values, Sūrikaisekikenkyūsho Kōkyūroku, 843, 162-170 (1993)
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