Euler-Stieltjes constants for the Rankin-Selberg L-function and weighted Selberg orthogonality

Abstract

Abstract. Let E be Galois extension of Q of finite degree and let π and π ′ be two irreducible automorphic unitary cuspidal representations of GLm(E A ) and GL m ′ (E A ), respectively. We prove an asymptotic formula for computation of coefficients γ π,π ′ (k) in the Laurent (Taylor) series expansion around s = 1 of the logarithmic derivative of the Rankin-Selberg L−function L(s, π × π ′ ) under assumption that at least one of representations π, π ′ is self-contragredient and show that coefficients γ π,π ′ (k) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π ′ is self-contragredient by a weaker one.

On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

“…If π v or π v ′ ramified, we can also write the local factors at ramified places v in the same form (2.1) with the convention that some of α π (v, j) and α π ′ (v, k) may be zero (see e.g. [27]).…”

“…Results related to the Hurwitz zeta function are given in [3], those for the Dedekind zeta function in [15,33], for the general setting of a non-co-compact Fuchsian group with unitary representation in [2], for a class of functions possessing an Euler product representation in [14], for a subclass S ♭ of the Selberg class in [39], for the extended Selberg class in [17] and for the Rankin-Selberg L−functions in [27,28]. Also, some investigations are done in the case of zeta functions with multiple variables, introducing multiple Stieltjes constants, for example, see [22,34].…”

On relations equivalent to the generalized Riemann hypothesis for the Selberg class

“…In [14] it was shown that the (finite) automorphic L-function L(s, π), defined by the product of its local factors (15) belongs to the class S ♯♭ . Moreover, for F (s) = L(s, π) (which is a function in S ♯♭ ) we have r = N, Q F = Q(π) 1/2 π −N/2 , λ j = 1/2, µ j = 1 2 k j (π), j = 1, ..., N and d F = N. Furthermore, when N = 1 and π is trivial, F (s) = L(s, π) reduces to the Riemann zeta function, hence, in this case m F = 1 and when N = 1 or π is not trivial, the function F (s) = L(s, π) is holomorphic at s = 1, hence m F = 0.…”

“…The coefficients γ F (k) are called the generalized Euler-Stieltjes constants of the second kind (see e.g. [15] for a more detailed explanation).…”

“…In [14] it was shown that the (finite) automorphic L-function L(s, π), defined by the product of its local factors (15) belongs to the class S ♯♭ . Moreover, for…”