SOME MEAN VALUE THEOREMS FOR THE RIEMANN ZETA-FUNCTION AND DIRICHLET L-FUNCTIONS D.A. KAPTAN, Y. KARABULUT, C.Y. YILDIRIM 1.

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1 SOME MEAN VAUE HEOREMS FOR HE RIEMANN ZEA-FUNCION AND DIRICHE -FUNCIONS D.A. KAPAN Y. KARABUU C.Y. YIDIRIM Dedicated to Professor Akio Fujii on the occasion of his retirement 1. INRODUCION he theory of the Riemann zeta-function ζs) and Dirichlet -functions s χ) abounds with unsolved problems. Chronoically the first of these now known as the Riemann Hypothesis RH) originated from Riemann s remark that it is very probable that all non-trivial zeros of ζs) lie on the line R s = 1 2. ater on Piltz conjectured the same for all of the functions s χ) GRH). he vertical distribution of the zeta zeros is the subject of Montgomery s pair correlation conjecture which can also be generalized for s χ). In this theory there are many other questions most of them still open about value distribution non-existence of linear relations among zeros of a function non-existence of common zeros of these functions etc. he results mentioned in this article may be seen as a first step in addressing some of these matters. We adopt the usual notation s = σ + it C with σ t R. he non-trivial zeros of ζs) will be denoted as ρ = β + iγ and those of s χ) as ρχ) = βχ) + iγχ). he parameter tends to. It is well-known that the number of zeros of ζs) in 0 < t is N ) = + O ) and for a primitive character χ to e the modulus q this number is N χ) = q + O q ). In this paper the e Dirichlet -functions are always those associated with primitive characters. he Gaussian sum associated with a character χmod q) is τχ) = q m=1 χm)e im q. For basic facts about the Riemann zeta-function and Dirichlet -functions we refer the reader to the books of Davenport [1] Montgomery and Vaughan [9] and itchmarsh [10] and for related material to Gonek [6]. Among the contributions of A. Fujii to this subject are the following results from [2] [5]: ζ ρ) 4π 2 0<γχ) iγ χ) 1) = 1 χ)χ 1)τχ) µq) φq) + 1 χ) iγχ) ψ) 1) = δq χ q ψ )1 ψχ)ψ 1)τψ) τχψ 0) φq ψ ) ψχ) on GRH fors χ))

2 2 where ψ is a primitive character to the modulus q ψ 3 and χ is a primitive character to the modulus q χ 1 χ ψ ψ 0 is the principal character mod q ψ and δq χ q ψ ) = 1 if q χ q ψ and = 0 otherwise. For the first formula here Fujii also gave the secondary terms and the error term both unconditionally and conditionally on RH. In fact Fujii proved more general versions and variations involving i γ K where K is an integer 2 and with certain weight functions in the summands) of these formulas. 2. SAEMEN OF RESUS he results announced here are generalizations of the above quoted results of Fujii and are all unconditional. For mean values over the zeros of the Riemann zeta-function we showed ζ j) ρ) = 1)j+1 ) j+1 + Oj j ) j 1) j + 1 where ζ j) s) is the j-th derivative of ζs). Assuming RH this reveals very clearly that there are points s on the critical line which are in particular zeros of ζs) at which the size of ζ j) s) is j. Next for χ a primitive character mod q 3 q ) A where A can be taken to be an arbitrarily large fixed number we have ) µq)χ 1)τχ)1 χ) ) ρ χ) 1 = 1 χ) A ) φq) + O e c and for j 1 j) ρ χ) = µq)χ 1)τχ) φq) j w=0 j! w! j v=w 1) v v w) 1 χ) q ) w v w)! + ) j)1 ) χ) + OAj e c where c is a non-effective constant depending on A and j. Using the bounds j) 1 χ) j q) j+1 j = ) ) j)1 χ) Aj q j+1 A j = ) 1 1 χ) A q 1 A these results imply for the range of q specified as above the simpler forms ρ χ) = )) 1 + O A i.e. the average value of ρ χ) is 1; 1 ρ χ) = µq)χ 1)τχ) 1 χ) N ) φq)

3 so that since φq) q q τχ) = q 1 χ) q the average value of ρ χ) gets closer to 0 with larger q; for j 1 we see that j) ρ χ)= 1)j+1 µq)χ 1)τχ)1 χ) φq) ) j 1 + O Aj if q is square-free and if q is not square-free then j) ρ χ) = ) j)1 χ) + OAj ) e c ) j)1 in which case since we do not know how close χ) can get to 0 we can only say j) ρ χ) Aj. For mean values over the zeros with ordinates in [0 ] of a Dirichlet -function s χ) where χ is a primitive character to the modulus q χ exp ) c c being an appropriate positive constant the averages of ζ µ) s) are ζρχ ) 1 ) = 1 χ) ζβ) ) β + O β 2 exp C ) ) and for µ 1 ζ µ) ρ χ ) = [ d )µ ds ]s=1 s χ) µ ) µ κ µ µ κ)! 1) j ζ κ) β) ) β ) µ κ j κ µ κ j)! β j+2 κ=0 j=0 + O exp C ) ) where β is the possible exceptional zero of s χ) and the terms involving it are present only if β exists) and C is some positive constant. For q satisfying q q) 4 all of the terms involving β can be absorbed into the error term. Next in the cases µ 1 we have for q ) A with any fixed A > 0 µ) ρ χ χ) = 1)µ+1 q ) µ+1 + O ) µ+ɛ ) µ + 1) for any fixed ɛ > 0. Now let ψ be a primitive character to the modulus q ψ and ψ 0 denote the principal character modulo q ψ. Assume that q = [q χ q ψ ] the least common multiple) satisfies q q) 4. We have ρχ ψ) 1 ) = δq χ q ψ ) ψ 1)τψ)τχψ 0)1 χψ) φq ψ ) 3 )) + 1 χψ)

4 4 For µ 1 and + O exp C q ψ ) ). µ) ρ χ ψ) = ψ 1)τψ)τχψ 0)1 χψ) φq ψ ) + O ) µ 1+ɛ) if q χ q ψ q ψ µ) ρ χ ψ) = 1) µ[ d ds )µ ] s=1 s ψχ) + O exp C q ψ ) ) if q χ q ψ. ) µ 3. REMARKS ABOU HE PROOFS he basic idea of the proofs is to see the sums to be evaluated as a sum of residues so we use f ρ) = 1 fs) g s) ds i R g c< γ< where ρ with imaginary part γ runs through the zeros of g and R is the rectangle having corners at a+ic a+i 1 a+i 1 a+ic with an appropriate c a constant) and so as to avoid the poles of the integrand). Using well-known estimates for ζs) s χ) and their derivatives the horizontal integrals and the integral on the vertical segment with real part a can be bounded easily and the main term is seen to come from the integral along the vertical segment with real part 1 a. Classical techniques and results of analytic number theory such as Perron s formula Dirichlet s hyperbola method partial summation the prime number theorem also for arithmetic progressions) the Pólya-Vinogradov inequality are employed in the calculations. he detailed version of the proofs are in [7] and [8]. References [1] H. Davenport Multiplicative number theory 3rd ed. revised by H.. Montgomery) Springer New York [2] A. Fujii Uniform distribution of the zeros of the Riemann zeta function and the mean value theorems of Dirichlet -functions Proc. Japan Acad. 63 Ser. A 1987) [3] A. Fujii Zeta zeros and Dirichlet -functions Proc. Japan Acad. 64 Ser. A 1988) [4] A. Fujii Some observations concerning the distribution of the zeros of the zeros of the zeta function II) Comment. Math. Univ. St. Pauli 40 No ) [5] A. Fujii On a conjecture of Shanks Proc. Japan Acad. 70 Ser. A 1994) [6] S.M. Gonek Mean values of the Riemann zeta-function and its derivatives Invent. Math ) [7] D.A. Kaptan Some mean values related to Dirichlet -functions M.Sc. hesis Bo gaziçi University S5

5 5 [8] Y. Karabulut Some mean value problems about Dirichlet -functions and the Riemann zetafunction M.Sc. hesis Bo gaziçi University S5 [9] H.. Montgomery and R.C. Vaughan Multiplicative number theory I. Classical theory Cambridge University Press [10] E.C. itchmarsh he theory of the Riemann zeta function 2nd ed. revised by D.R. Heath- Brown Oxford University Press Deniz Ali Kaptan Kaptan Central European University Department of Mathematics and its Applications Nador u Budapest Hungary Yunus Karabulut yunus.karabulut1@boun.edu.tr) Cem Yalçın Yıldırım yalciny@boun.edu.tr): Department of Mathematics Bo gaziçi University Bebek Istanbul urkey