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We consider the first moment of the value of automorphic L-functions at an arbitrary point on the critical line, a sum over primitive forms weighted by their Petersson's norm. In this paper, we obtain an asymptotic formula for it when weight k is an even integer satisfying 0 k 12 or k = 14 and level is p α , where p is a prime number. This formula yields a lower bound of the number of primitive forms, whose L-functions do not vanish at that point.

Background We have recently found that exposure to acute restraint stress suppresses antigen-specific antibody production, including IgE, in a murine model of allergic rhinitis. Although age-related alterations in immune responses are known, it remains unclear whether aging modulates the antibody production under stressful conditions. In this study, we set out to determine the effects of aging on antibody production under acute restraint stress in mice. Methods Both young and aged CBA/J mice were repeatedly sensitized intranasally with phospholipase A2 (PLA2) without adjuvants. Restraint stress was applied using uniform cylinders once a week for a continuous 8 h period, on 5 occasions in total. Blood samples were taken at 0, 20 and 30 days after primary sensitization, and production of PLA2-specific antibodies and levels of IL-4, IFN-γ, IL-10 and IL-1β in sera were determined by ELISA. Results Repeated intranasal sensitization with PLA2 induced PLA2-specific IgE, IgG1 and IgG2a production in aged mice. We found that exposure to restraint stress significantly inhibited production of PLA2-specific IgE, IgG1 and IgG2a in aged mice. In addition, antibody production under restraint stress decreased significantly in aged mice when compared with young mice. No IL-4, IFN-γ, IL-10 or IL-1β were detected in sera from non-stressed or stressed aged mice. Conclusions Aging exacerbates the immunosuppressive role of acute restraint stress in antigen-specific antibody production in mice.

We consider an effective lower bound of the Siegel-Tatuzawa type for general L-functions with three standard assumptions. We further assume three hypotheses in this paper that are essential in developing our argument.Under these assumptions and hypotheses, we prove a theorem of Siegel-Tatuzawa type for general L-functions. In particular, we prove such a theorem for symmetric power L-functions under certain assumptions.

Background: Exposure to acute stressors modulates both innate and acquired immune function. However, little is known about whether stress has the potential to modulate the pathogenesis of allergic rhinitis. Objectives: To determine the effects of acute restraint stress on the initiation of allergic rhinitis in a murine model. Methods: CBA/J mice were repeatedly intranasally sensitized with phospholipase A2 (PLA2) from honeybee venom without adjuvant. Restraint stress was applied using uniform cylinders once a week for a continuous 8-hour period, on five occasions in total. Production of PLA2-specific antibodies and degree of nasal and blood eosinophilia were compared between stressed and control mice. Results: Repeated intranasal sensitization with PLA2 induced PLA2-specific IgE and marked eosinophilia in both the nose and blood in CBA/J mice. Exposure to restraint stress significantly inhibited production of PLA2-specific IgE, IgG1 and IgG2a. Conversely, the stress exerted no significant effect on eosinophilia. Conclusions: Exposure to acute restraint stress inhibits antigen-specific antibody production, but not local or systemic eosinophilia. The results of this study suggest that acute stress has the potential to modulate the initiation of allergic rhinitis.

Rankin obtained the asymptotic formula for the sum of coefficients of Rankin-Selberg $L$ -series associated with a cusp form and a trivial character. $\mathrm{Ivi}\acute{\mathrm{c}}-$ Matsumoto-Tanigawa studied the error term in it by using a mean value formula which is yielded from the Voronoï formula of the Riesz mean. In this paper, we consider more general Rankin-Selberg $L$ -series. It is associated with two cusp forms and a nontrivial character mod $d$ . $\mathrm{Ivi}\acute{\mathrm{c}}$ -Matsumoto-Tanigawa's method cannot be applied directly to our case. We consider the sum of coefficients of twisted Rankin-Selberg $L$ -series by a modification of their method.

We study where c n s are the coefficients of the Rankin-Selberg series, p is an odd prime, r is a natural number, and a is also a natural number satisfying (a, p) = 1. For any natural number d, we know the asymptotic formula for , where χ is a primitive Dirichlet character mod d. This is obtained by using the Voronoiformula of the Riesz-mean . In particular, in case d = p r,the fourth power of the Gauss sum appears in that Voronoiformula. We consider the sum over all characters mod pr, then the fourth power of the Gauss sum produces the hyper-Kloosterman sum. Hence, applying the results of Deligne and Weinstein, we can estimate the error term in the asymptotic formula for .