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On the modified Li criterion for a certain class of $$L$$-functions. (English) Zbl 1347.11063
The authors consider a class $$\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$$ of $$L$$-functions that contains the Selberg class, the class of all automorphic $$L$$-functions and the Rankin-Selberg $$L$$-functions, as well as products of suitable shifts of those functions. They prove the generalized Li criterion for zero-free regions of functions belonging to the class $$\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$$, derive an arithmetic formula for $$\tau$$-Li coefficients and conduct numerical investigation of $$\tau$$-Li coefficients for a certain product of shifts of the Riemann zeta function.
More precisely, for real numbers $$\sigma_0$$ and $$\sigma_1$$ such that $$\sigma_0 \geq \sigma_1>0$$, the class $$\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$$ is the class of functions $$F$$ satisfying the following four axioms:
$$\bullet$$
(Dirichlet series) The function $$F$$ possesses a Dirichlet series representation that converges absolutely for $$\mathrm{Re} (s) >\sigma_0.$$
$$\bullet$$
(Analytic continuation) There exist finitely many non-negative integers $$m_1, \dots, m_N$$ and complex numbers $$s_1, \dots, s_N$$ such that the function $$\prod\limits_{i=1}^{N} (s-s_i)^{m_i}F(s)$$ is an entire function of finite order.
$$\bullet$$
(Functional equation) The function $$F$$ satisfies the functional equation $$\xi_{F}(s)=\omega \overline{\xi_{F}(\sigma_1-\bar{s})},$$ where the completed function $$\xi_F$$ is defined as $\begin{split} \xi _{F}(s)=&F(s) Q_{F}^{s}\prod_{j=1}^{r}\Gamma (\lambda _{j}s+\mu _{j})\prod\limits_{i=1}^{2M + \delta(\sigma_1)} (s-s_i)^{m_i} \\ &\prod_{i=2M+1 + \delta(\sigma_1)}^{N}(s-s_i)^{m_i} (\sigma_1 - s-\overline{s_i})^{m_i}, \end{split}$ where $$\left| \omega \right| =1$$, $$Q_{F}>0$$, $$r\geq 0$$, $$\lambda _{j}>0$$, $$\mu_j\in\mathbb C$$, $$j=1,\ldots ,r$$, and $$\Gamma$$ is the Euler Gamma function. It is assumed that the poles of $$F$$ are arranged so that the first $$2M+\delta(\sigma_1)$$ poles ($$0\leq 2M + \delta(\sigma_1) \leq N$$) are such that $$s_{2j-1} + \overline{s}_{2j} = \sigma_1$$, for $$j=1,\ldots, M$$, and $$\delta(\sigma_1) = 1$$ if $$\sigma_1/2$$ is a pole of $$F$$ in which case $$s_{2M+\delta(\sigma_1)} = \sigma_1/2$$; otherwise $$\delta(\sigma_1)=0$$.
$$\bullet$$
(Euler sum) The logarithmic derivative of the function $$F$$ possesses a Dirichlet series representation
converging absolutely for $$\mathrm{Re} (s)>\sigma_0$$.
\smallskip
The non-trivial zeros of $$F$$ are defined to be the zeros of the completed function $$\xi_F$$. The set of non-trivial zeros of $$F(s)$$ is denoted by $$Z(F)$$. By the functional equation and the Euler sum, all those zeros lie in the critical strip $$\sigma_1-\sigma_0\leq \mathrm{Re} (s) \leq \sigma_0$$. Let $$\tau\in[\sigma_1,+\infty)$$. For an arbitrary positive integer $$n$$, the $$n$$th $$\tau$$-Li coefficient associated to the $$F\in\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$$ is defined as $\lambda_{F}(n,\tau)=\left.\sum_{\rho\in Z(F)}\right.\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n\right),$ where the sum is taken in the sense of the limit $$\lim\limits_{T \to \infty} \sum\limits_{|\mathrm{Im}(\rho)| \leq T}$$. The main result of the paper is the following Li-type criterion. Let $$0,\tau\notin Z(F)$$. The next two statements are equivalent
(i)
$$\sigma_1-\frac{\tau}{2}\leq \mathrm{Re} (\rho)\leq\frac{\tau}{2}$$ for every $$\rho\in Z(F)$$,
(ii)
$$\mathrm{Re} (\lambda_F(n,\tau))\geq 0$$ for every positive integer $$n$$.

MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 11M41 Other Dirichlet series and zeta functions
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References:
 [1] Avdispahić, M.; Smajlović, L., ϕ-variation and Barner-Weil formula, Math. Balkanica, 17, 3-4, 267-289, (2003) · Zbl 1064.42004 [2] Avdispahić, M.; Smajlović, L., Explicit formula for a fundamental class of functions, Bull. Belg. Math. Soc. Simon Stevin, 12, 569-587, (2005) · Zbl 1210.11097 [3] Avdispahić, M.; Smajlović, L., A note on Weil’s explicit formula, (Krehnnikov, A. Y.; Rakić, Z.; Volovich, I. V., p-adic Mathematical Physics: 2nd International Conference on p-adic Mathematical Physics, (2006), American Institute of Physics New York), 312-319 · Zbl 1152.11339 [4] Bombieri, E.; Lagarias, J. C., Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory, 77, 274-287, (1999) · Zbl 0972.11079 [5] Bucur, A.; Ernvall-Hytönen, A.-M.; Odžak, A.; Roditty-Gershon, E.; Smajlović, L., On τ-Li coefficients for Rankin-Selberg L-functions, (Bucur, A.; etal., Women in Numbers Europe, Association for Women in Mathematics Series, vol. 2, (2015), Springer International Publishing Switzerland) · Zbl 1383.11065 [6] Coffey, M., Toward verification of the Riemann hypothesis: application of the Li criterion, Math. Phys. Anal. Geom., 8, 211-255, (2005) · Zbl 1097.11042 [7] Droll, A. D., Variations of Li’s criterion for an extension of the Selberg class, (2012), Queen’s University Ontario Canada, PhD thesis [8] Freitas, P., A Li-type criterion for zero-free half-planes of Riemann’s zeta function, J. Lond. Math. Soc., 73, 399-414, (2006) · Zbl 1102.11046 [9] Jorgenson, J.; Lang, S., Basic analysis of regularized products and series, Lecture Notes in Math., vol. 1564, (1993), Springer-Verlag Berlin, Heidelberg · Zbl 0788.30003 [10] Jorgenson, J.; Lang, S., Explicit formulas for regularized products and series, Lecture Notes in Math., vol. 1593, (1994), Springer-Verlag Berlin, Heidelberg · Zbl 0828.11043 [11] Kaczorowski, J.; Perelli, A., On the structure of the Selberg class, I: $$0 \leq d \leq 1$$, Acta Math., 182, 207-241, (1999) · Zbl 1126.11335 [12] Karatsuba, A. A.; Korolev, M. A., The argument of the Riemann zeta function, Russian Math. Surveys, 60, 3, 433-488, (2005) · Zbl 1116.11070 [13] Lagarias, J. C., Li’s coefficients for automorphic L-functions, Ann. Inst. Fourier, 57, 1689-1740, (2007) · Zbl 1216.11078 [14] Li, X.-J., The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory, 65, 325-333, (1997) · Zbl 0884.11036 [15] Li, X.-J., Explicit formulas for Dirichlet and Hecke L-functions, Illinois J. Math., 48, 491-503, (2004) · Zbl 1061.11048 [16] Maślanka, K., Li’s criterion for the Riemann hypothesis - numerical approach, Opscula Math., 24, 103-114, (2004) · Zbl 1136.11319 [17] Mazhouda, K., On the τ-Li coefficients for automorphic L-functions, Rocky Mountain J. Math., (2015), in press [18] Odlyzko, A., Tables of zeros of the Riemann zeta function · Zbl 0706.11047 [19] Odžak, A.; Smajlović, L., On Li’s coefficients for the Rankin-Selberg L-functions, Ramanujan J., 21, 303-334, (2010) · Zbl 1248.11036 [20] Odžak, A.; Smajlović, L., On asymptotic behavior of generalized Li coefficients in the Selberg class, J. Number Theory, 131, 519-535, (2011) · Zbl 1257.11082 [21] Omar, S.; Ouni, R.; Mazhouda, K., On the zeros of Dirichlet L-functions, LMS J. Comput. Math., 14, 140-154, (2011) · Zbl 1294.11144 [22] Omar, S.; Ouni, R.; Mazhouda, K., On the Li coefficients for the Hecke L-functions, Math. Phys. Anal. Geom., 17, 67-81, (2014) · Zbl 1356.11060 [23] Selberg, A., Old and new conjectures and results about a class of Dirichlet series, (Bombieri, E.; etal., Proc. Amalfi Conf. Analytic Number Theory, Universitia di Salerno, (1992)), 367-385 · Zbl 0787.11037 [24] Smajlović, L., On Li’s criterion for the Riemann hypothesis for the Selberg class, J. Number Theory, 130, 828-851, (2010) · Zbl 1188.11046 [25] Voros, A., Sharpening of Li’s criterion for the Riemann hypothesis, Math. Phys. Anal. Geom., 9, 53-63, (2006) · Zbl 1181.11055
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