THE CONFERENCE THEORY OF THE RIEMANN ZETA AND ALLIED FUNCTIONS AT OBERWOLFACH

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1 39 Proc. Sci. Seminar Faculty of Physics and Mathematics, Šiauliai University, 5, 22, THE CONFERENCE THEORY OF THE RIEMANN ZETA AND ALLIED FUNCTIONS AT OBERWOLFACH Antanas LAURINČIKAS Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 26, Vilnius, Lithuania. Faculty of Physics and Mathematics, Šiauliai University, Vytauto 84, 54, Šiauliai, Lithuania. Abstract. In the paper an overview of the conference on the analytic number theory at Oberwolfach is presented. Key words: divisor problem, limit theorem, Oberwolfach, Riemann hypothesis, Riemann zeta-function. The Oberwolfach Institute of Mathematics was found in September 944 as an investition in scientific research to try to ensure a victory. The principal organizer of this institute was Professor Wilhelm Süss. He was a geometer, a student of Professor L. Bieberbach. At the end of the war only few mathematicians worked in a mathematical institute, together with members of their families about twenty people had an accomodation in Oberwolfach. After the war the Institute of Mathematics continued its existence. The director W. Süss wanted to make an international institute,and his efforts were successful. The institute became a place for meetings between Germain mathematicians and their foreign colleagues. At first the contacts were reestablished between Germain and French mathematicians. The first organized meeting in Oberwolfach took place in 949. During three or five meetings were held every year. The famous mathematicians J. Dieudonné, J.-P. Serre, R. Thom, K. Leichtweiss were among participants of the first meetings. The number of meetings increased each year. In 96 after finansial support from the Fritz Thyssen Foundation the number of conferences reached twenty per year. In 967 the guesthouse was built, and Oberwolfach became the world s mathematical meeting place. Today one conference per week almost every week of the year is held in Oberwolfach. The total number of meetings organized in Oberwolfach exceeds 3. On September 6 22, 2 the meeting Theory of the Riemann zeta and allied functions was held in Oberwolfach. Professors M. N. Huxley (Cardiff), M. Jutila (Turku) and Y. Motohashi (Tokyo) organized this meeting. More than 3 participants gave their talks on analytic number theory. Here we would like to present an overview of some talks of the conference.

2 4 The conference Theory of the Riemann zeta... E. Bombieri (Princeton) discussed a variational approach to Weil s explicit formula. Let f(x) C (R + ) be a smooth complex-valued function with compact support in R +. Define f (x) = ( ) x f, x and let f(x) = f(x)x s dx denote the Mellin transform of f(x). Then the explicit formula has the following form: f(ρ) = f(x)dx + f (x)dx W v (f), ρ v where the first sum ranges over all complex zeros of the Riemann zeta-function ζ(s), v runs over the valuators {R, 2, 3, 5,...} and W p (f) = (log p) { f(p m ) + f (p m ) }, m= ( W R (f) = (log 4π + γ )f() + f(x) + f (x) 2 ) xdx x f() x 2 ( ( ( )) ) f() Γ w = log π Re f(w)dw. 2π Γ 2 ( 2 ) We recall that the Riemann hypothesis (RH) asserts that all complex zeros of ζ(s) lie on the critical line. In Bombieri s talk some equivalents of RH were presented. Theorem. RH holds if and only if g(ρ) g( ρ) > for every complex-valued function g(x) C (R + ), g(x). Theorem 2. Let T [f] be a linear functional defined by T [f] = ρ f(x)dx + f (x)dx v W v (f) on the space C (R + ) of complex-valued smooth functions with compact support in R +. Then T [f] = T [f ] = f(ρ), ρ

3 A. Laurinčikas 4 where the sum runs over all complex zeros of ζ(s). Moreover, RH is equivalent to the inequality T [f f ] on C (R + ), and = f. E. Bombieri also stated two problems. Problem A. Let E be a finite union of intervals on R +. To minimize the functional T [f f ] in the unit sphere of the space L 2 (E) of functions f with compact support in E, and with norm f 2 = f(x) 2 dx. E Problem B. Let E = [M, M], M >. To minimize the functional T [f f ] in the unit sphere of L 2 (E) with norm f 2 M = M f(x) 2 dx + M M f(x) 2 dx, where M = 2 D, M ( ) d D = x. dx Theorem 3. The infimum of the functional T [f f ] in the unit sphere of the space L 2 (E) of L 2 -functions with compact support in E is obtained. T. Meurman (Turku) gave a talk on the additive divisor problem. Let, as usual, d(n) = d n. Further, D(N; f) = n N d(n)d(n + f), N M(N; f) = (a f log x log(x + f) + b f log x(x + f) + c f )dx and E(N; f) = D(N; f) M(N; f). A problem is to evaluate E 2 (N; f). f F

4 42 The conference Theory of the Riemann zeta... Theorem 4. For N, F and ε > we have E 2 ( 4 3 ) (N; f) ε N 3 F 3 + NF + N 2 F 2 (N + F ) ε. f F The bound of the theorem is nontrivial for F N 3 and new for N 2 F N 3. Theorem 5. The estimate E(N; f) ( N(N + f) ) 3 N ε + ( N(N + f) ) 4 min ( N 4 ; f 8 + α 2 ) N ε holds with α J. Brüdern (Stuttgart) considered the representation of primes as sum of k-th powers: p = x k x k s. () Let For example, P (k) = min{s : () has solutions for infinitely many p}. p = x 2 + y 2 Therefore in this case P (2) = 2. Also, p 3(mod4). p = x 3 + 2y 3 for infinitely many p, thus we have P (3) = 3. It is expected that P (k) 3 for all k. It is known that P (k) k log k + O(k log log k). 2 Theorem 6. Assume GRH. Then P (k) 8 3 k. Moreover, for s > 8 3 k, # { p X : p = x k x k } s X θ with { θ = exp 2s }. k M. N. Huxley discussed exponential sums and their applications. For the Riemann zeta-function he obtained a bound ( ) ζ 2 + it t (log T ) A. Note that = Similarly, in the circle problem the formula πr 2 + O ( R 3 28 (log R) B )

5 A. Laurinčikas 43 was proved, and for the divisor problem the following result was obtained: d(m) = N log N + (2γ )N + O ( ) N m N The author presented in his talk a limit theorem for the Riemann zeta-function in the space of continuous functions. Let C denote the complex plane, and let C = C { } be the Riemann sphere with a metric d given by the formulae 2 s s 2 d(s, s 2 ) = + s 2 + s 2, d(s, ) = 2, d(, ) =, 2 + s 2 s, s, s 2 C. This metric is compatible with the topology of C. Let C(R) = C(R, C ) denote the space of continuous functions f : R C equipped with the topology of uniform convergence on compacta. In this topology, a sequence {f n, f n C(R)} converges to the function f C(R) if uniformly in t on compact subsets of R. Let γ be the unit circle on C, and d(f n (t), f(t)), n, Ω = γ p, p where γ p = γ for each prime p. With product topology and pointwise multiplication Ω is a compact topological Abelian group. Therefore there exists the probability Haar measure m H on (Ω, B(Ω)), where B(S) stands for the class of Borel sets of the space S. This gives a probability space (Ω, B(Ω), m H ). Let ω(p) be the projection of ω Ω to γ p. Moreover, let, for m N, ω(m) = ω α (p). p α m Denote by d a (m) the coefficients of the Dirichlet series expansion of ζ a (s) in the half-plane σ >. Then it can be proved that d κt (m)ω(m) m σ T +it m T for almost all ω Ω converges uniformly in t on compact subsets of R to some function S(t, ω) as T. Therefore, S(t, ω) is a C(R)-valued random element defined on the probability space (Ω, B(Ω), m H ). Denote by P S the distribution of S(t, ω), i. e. P S (A) = m H ( ω Ω : S(t, ω) A ), A B(C(R)). Theorem 6. Let θ > 2 2 be fixed, κ T = ( 2 log log T ) and σ T = 2 + θ(log log T ) 3 2. log T Then under RH the probability measure T meas{ τ [, T ] : ζ κ ( T σ T + it + iτ ) A }, converges weakly to P S as T. A B(C(R)),

6 44 The conference Theory of the Riemann zeta... Konferencija Rymano dzeta ir kitu giminingu funkciju teorija Oberwolfache A. Laurinčikas Straipsnyje pateikiama analizinės skaičiu teorijos konferencijos Oberwolfache apžvalga. Rankraštis gautas