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ON SOME REASONS FOR DOUBTING THE RIEMANN HYPOTHESISAleksandar Ivi´cAbstract. Several arguments against the truth of the Riemann hypothesis areextensively discussed.These include the Lehmer phenomenon, the Davenport–Heilbronn zeta-function,large and mean values of 

|_( 1

2 + it)| on the critical line, and zeros of a class of convolution functions closely related to _( 1

2 + it). Thefirst two topics are classical, and the remaining ones are connected with theauthor’s recent research. By means of convolution functions a conditional disproof of the Riemann hypothesis isgiven.

0. Foreword (“Audiatur et altera pars”) This is the unabridged version of the work that was presented at the BordeauxConference in honourof the Prime Number Theorem Centenary, Bordeaux, January 26, 1996 and laterduring the 39th TaniguchiInternational Symposium on Mathematics “ Analytic Number Theory ”, May 13-17,1996 in Kyoto and itsforum, May 20-24, 1996. The abridged printed version, with a somewhat differenttitle, is [62]. The multiplicitiesof zeros are treated in [64]. A plausible conjecture for the coefficients of the mainterm in theasymptotic formula for the 2k-th moment of |_( 1

2 + it)| (see (4.1)–(4.2)) is given in [67].In the years that have passed after the writing of the first version of this paper, itappears that thesubject of the Riemann Hypothesis has only gained in interest and importance. Thisseem particularly true in

view of the Clay Mathematical Institute prize of one million dollars for the proof of the Riemann Hypothesis,which is called as one of the mathematical “Problems of the Millenium”. Acomprehensive account is tobe found in E. Bombieri’s paper [65]. It is the author’s belief that the present workcan still be of interest,especially since the Riemann Hypothesis may be still very far from being settled.Inasmuch the RiemannHypothesis is commonly believed to be true, and for several valid reasons, I feelthat the arguments thatdisfavour it should also be pointed out.One of the reasons that the original work had to be shortened and revised beforebeing published is the

remark that “The Riemann hypothesis is in the process of being proved” bypowerful methods from Randommatrix theory (see e.g., B. Conrey’s survey article [66]). Random matrix theory hasundisputably found itsplace in the theory of _(s) and allied functions (op. cit. [66], [67]). However, almostten years have passedsince its advent, but the Riemann hypothesis seems as distant now as it was then.