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The Siegel-Walfisz theorem for Rankin-Selberg \(L\)-functions associated with two cusp forms. (English) Zbl 0977.11022

Let \(S_k(\Gamma)\) be the space of cusp forms of weight \(k\) for \(\Gamma=SL_2(\mathbb Z)\). Let \(f\in S_k(\Gamma)\) and \(g\in S_l(\Gamma)\) be normalized Hecke eigen forms and let \(L_{f \otimes g}(s,\chi)\) be the twisted Rankin-Selberg \(L\)-function associated with \(f\) and \(g\), where \(\chi\) is a primitive Dirichlet character modulo \(d\). In this paper, the author obtains a zero-free region of \(L_{f \otimes g}(s,\chi)\) except for at most one exceptional zero (the Siegel zero). Next, for real primitive characters \(\chi\) modulo \(d\geq 2\), it is proved an analogue of Siegel’s theorem for a zero-free region on the real axis for \(L_{f \otimes g}(s,\chi)\).
The main result of this paper is an analogue of the Siegel-Walfisz prime number theorem for the product of the Fourier coefficients of two cusp forms. Assume \(f\neq g\), and let \(a\) be a positive integer with \((a,d)=1\) and \(M\) be a positive number. Then there exists a positive constant \(c=c(M)\) which depends on \(M\) such that \[ \sum_{\substack{ p:\text{prime}\leq x\\ p\equiv a \pmod d }} a_pb_p =O(x^{(k+l)/2}\exp(-c\sqrt{\log x})) \] for \(d \leq (\log x)^M\), where \(a_n\) and \(b_n\) are \(n\)-th Fourier coefficients of \(f\) and \(g\) respectively. A. Perelli [Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 405-410 (1985; Zbl 0589.10029)] obtained a related result for the case of \(f=g\).

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
11N05 Distribution of primes
11F11 Holomorphic modular forms of integral weight

Citations:

Zbl 0589.10029
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