An Extension of the Linnik Phenomenon

Abstract

This work is a continuation of [9] but can be read independently. We discuss the extension of the Linnik phenomenon to automorphic L-functions associated with cusp forms, focusing our attention on the real analytic situation, as the holomorphic case is settled in [9]. Our main assertion, which is given at the end of the fifth section, reveals that the repelling effect of exceptional zeros of Dirichlet L-functions should be felt not only by those L-functions themselves but also by automorphic L-functions. We stress that constants, including those implicit, are all universal and effectively computable, unless otherwise stated.

On sums of Hecke-Maass eigenvalues squared over primes in short intervals

“…Our argument to prove (1.12) is similar, to an extent, to that of our recent work [15], though the present work can be read independently. Thus, we rely on the theory of symmetric power L-functions and on our observation [12, Section 2, 13, Section 1.4] concerning the arithmetical nature of the optimal weights in the general Λ 2 -sieve mechanism.…”

“…(ii) It might be worth remarking that one may avoid appealing to (1.11), since it is possible to prove an analogue of the Deuring-Heilbronn-Linnik phenomenon for the function L(s; V × V ) itself, by following the argument of [15,Part II]. Namely, one may allow the possibility of the existence of the Landau-Siegel zero for L(s; V × V ), as it should repel all other zeros towards the left of the line Re s = 1 deeper than (1.11).…”

“…Nonetheless, the employment of (2.9) in place of (2.4) is never trivial itself. The use of the factor τ (2) V ×V (n) is deliberately made in order to exploit the Rankin convolution, which is not necessary when dealing with holomorphic cusp forms because of the validity of the Ramanujan conjecture; see [15,Part I]. We stress that our argument should generalize to the treatment of sums of |τ V (p)| 2 over primes in short intervals, provided the holomorphy, the polynomial growth and the zero-free region of the type (2.12) are all available for L(s; sym 2 V × sym 2 V ); as to the last requirement, an appropriate extension of (ii) above will work.…”