## 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

Zbl 0589.10029
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