# Conference "Zeta Functions 5"

## December 1 - 5, 2014, Moscow, Russia

**Siegfried Boecherer (Mannheim) : On Dirichlet series of Rankin type for noncuspidal Siegel modular forms**

There are two methods to overcome the problem of non-rapid decay in the Rankin-Selberg method: Truncation and differential operators. We report on the application of the latter method for Siegel modular forms. The case of ordinary Rankin-convolutions a la Andrianov-Kalinin was treated in a joint paper with F.Chiera (2008). Now we deal with the more complicated case of Dirichlet series buildt from scalar products of Fourier-Jacobi-coefficients (Dirichlet series of Kohnen-Skoruppa-Yamazaki). This is ongoing joint work with S.Das (Bangalore).

**Florent Demeslay (Caen) : A class formula for L-series in positive characteristic**

Showing the analogy with the case of number fields, we state a class formula for function fields in positive characteristic, generalizing a result of Taelman. Then, we apply this class formula to two kinds of*L*-values.

**Aryan Farzad (Lethbridge) : On Binary and Quadratic Divisor Problem**

Let*d(n)=Σ*. This is known as divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum_{d|n}1

*Σ*_{an-m=h}d(n)d(m)f(an,m)

*f*is a smooth function which is supported on*[x, 2x]*x*[x,2x]*and oscillates mildly. In 1993, Duke, Friedlander, and Iwaniec proved that*Σ*_{an-m=h}d(n)d(m)f(an,m)=Main term(x)+ O(x^{0.75})

*O(x*, and conditionally, under the assumption of the Ramanujan-Petersson conjecture, to^{0.61})*O(x*. We will also give some new results on shifted convolution sums of functions coming from Fourier coefficients of modular forms.^{0.5})

**Daniel Fiorilli (Ottawa) : Primes in arithmetic progressions**

I would like to discuss primes congruent to*a*modulo*q*, with*a*fixed and on average over*q*. My goal is to show that Vaughan's approximation is superior to the usual approximation for this quantity, and that it has a bias towards certain values of*a*.

**Sergey Gorchinskiy (Moscow) : Higher Contou-Carrere symbol**

The talk is based on a joint work with D. Osipov. We define a higher-dimensional generalization of the Contou-Carrere symbol and discuss its universal property.

**Emmanuel Hallouin (Toulouse) : Graph based strategy to exhibit good recursive towers**

We describe a way to find good recursive towers based over graph theory. Despite the fact that this method do not lead, up to now, to a new good tower, we think that this method is a step toward a better understanding of recursive towers.

**Harald Helfgott (Paris) : The ternary Goldbach conjecture**

The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible*(C>10*. (Works by Ramare and Tao solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas of the proof.^{1300})

**Marc Hindry (Paris) :**

T.B.A.

**Alexander Ivanov (Munich) : Anabelian properties of arithmetic curves**

he Isom-conjecture of Grothendieck concerns the fact, that the isomorphism class of certain types of varieties is uniquely determined by their étale fundamental groups. While in the case of affine curves over a finite field the Isom-conjecture was solved by Tamagawa, up to now almost nothing is known for arithmetic curves. This is mainly because we lack an analogue of Lefschetz's fixed point theorem but also because the fundamental group of an arithmetic curve is still poorly understood. We discuss a new method to overcome at least the first difficulty, i.e., how to reconstruct the decomposition subgroups of points inside the fundamental group (assuming some still unknown properties of it), using some additional data and the Tsfasman-Vladut theorem, but avoiding any cohomology theory.

**Florent Jouve (Orsay) : Chebychev's bias for elliptic curves over function fields**

Chebychev was the first to point out that for "most" real numbers x the number of prime numbers up to x and congruent to 3 mod 4 exceeds that of primes up to x and congruent to 1 mod 4. This kind of phenomenon and its generalizations have in the recent years been called Chebychev's bias. In the 1990's Rubinstein and Sarnak gave a general framework to study that question. They notably emphasized the relevance of related Dirichlet L-functions and remark that much can be deduced about Chebychev's bias if one is willing to to believe that both the Riemann Hypothesis and a strong form of simplicity of the zeros holds for these*L*-functions. The aim of the talk is to present an analogous study in the setting of elliptic curves over the rational function field*F*. Given such an elliptic curve_{q}(t)*E/F*we will see how to study sums of type_{q}(t)*Σ*where_{deg v < x}cosθ_{v}*v*runs over the places of good reduction of*E/F*and_{q}(t)*±θ*are the arguments to the inverse roots of the numerator of the zeta function of the reduced elliptic curve_{v}*E*over the residual field at_{v}*v*. An important feature of this geometric setting is that the conjectures on zeros of L-functions in the classical case can often be replaced by theorems. (Joint work with B. Cha and D. Fiorilli.)

**Satoshi Kondo (Moscow) : On curves of genus***1*or*2*over functions fields with large rational*K*group_{2}

Joint work with Masataka Chida and Takuya Yamauchi. We showed that the rank of the rational*K*group of a curve of '_{2}*GL(2)*-type' over a function field was bounded below by the order of pole at certain integer point of the*L*-function. In the talk, we provide some explicit examples of genus*1*or*2*curves where the order is arbitrarily large. Thus we obtain curves with very large rational*K*groups._{2}

**Maxim Korolev (Moscow) : On the large values of modulus of the Riemann zeta function on the critical line**

n the talk, we will discuss some problems concerning the existence of very large values of real and imaginary part of the reimann zeta-function on very short intervals of critical line. In particular, it appears that for any large constant*A>0*the maximum of absolute value of*ζ(0.5+it)*on each interval*(T,T+H)*is greater than*A*when*H*is greater than triple logarithm of*T*. All these results are based on the Riemann hypothesis.

**Stéphane Louboutin (Marseille) : Applications of Stechkin's Lemma to lower bounds for***L(1,χ)*and Siegel's zeros of Dedekind Zeta functions

T.B.A.

**Yuri Matiyasevich (Saint-Petersburg) : Studying the zeta function via certain determinants**

n 2007 the speaker reformulated the Riemann Hypothesis as statements about the eigenvalues of certain Hankel matrices, entries of which are defined via the Taylor series coefficients of the zeta function. Numerical calculations revealed some very interesting visual patterns in the behavior of these eigenvalues. Recently computations have been extended and performed on more powerful computers. This led to new conjectures about the finer structure of the eigenvalues and eigenvectors and to conjectures that are (formally) stronger than RH. At the previous Zeta'2012 meeting the speaker presented another approach to studding the zeta function via determinants. A short survey of new discoveries along this line of research will be presented as well.

**Baptiste Morin (Bordeaux) :**

The goal of this talk is to give a conjectural description of leading Taylor coefficients of zeta functions in terms of finitely generated cohomology groups. If time permits, we will try to make this description explicit in special cases. This is joint work with Matthias Flach.

**Vésale Nicolas (Heidelberg) : Fine Selmer groups and the Equivariant Main Conjecture for elliptic curves**

T.B.A.

**Denis Osipov (Moscow) : The two-dimensional Contou-Carrere symbol and algebraic K-theory**

One-dimensional and two-dimensional Contou-Carrere symbols are natural deformations of usual and two-dimensional tame symbols correspondingly. One-dimensional Contou-Carrere symbol was first studied by C. Contou-Carrere and P. Deligne. I will give several constructions of two-dimensional Contou-Carrere symbol, will explain its connection with two-dimensional residue of differential forms and two-dimensional class field theory. I will point out connections with algebraic K-theory. Finally, I explain reciprocity laws for this sumbol on algebraic surfaces. The talk is based on joint results with X. Zhu which are contained in the paper: arXiv:1305.6032.

**Alexei Pantchichkine (Grenoble) :***p*-adic Zeta Functions and Quasimodular forms

A new method of constructing p-adic zeta-functios is described using quasi-modular forms and their Fourier coefficients.

**Marc Perret (Toulouse) : Weil bound of higher order**

We describe an euclidean process which permits to recover easily the Weil and Ihara bounds on the number of points on cuvres over finite fields. We show how this process leads to new bounds.

**Irina Rezvyakova (Moscow) : An additive problem with the Fourier coefficients of holomorphic non-cusp forms**

An additive problem is a problem of evaluating the following sum*S=Σ*_{an-bm=l,n≤N}a(n)a(m),*N*is a large number,*a(n)*are complex numbers, and*a,b,l*are positive integers, which can slightly grow with*N*. This type of problem arises in many situations. For example, an asymptotic formula (or a good estimate) for the sum*S*should be established, if one wants to obtain an asymptotic formula for the mean-value (on the critical line) of*L*-function, which corresponds to a Dirichlet series with the given coefficients*a(n)*. If the coefficients*a(n)*satisfy the Ramanujan-Petersson conjecture (namely, if the estimate*a(n) <<*is valid), then we have the trivial estimate_{ε}n^{ε}*S << N*, which is though not usually enough. We shall discuss the Kloosterman's variant of the circle method which allows to obtain an asymptotic formula (or a good estimate) for the sum^{1+ε}*S*with the Fourier coefficients of non-cusp (or cusp) automorphic forms. As an example, we consider the details of the method for the coefficients of Dirichlet series attached to Hecke*L*-function with real (or complex) ideal class group character.

**Sergey Rybakov (Moscow) : Coherent de Rham-Witt modules**

T.B.A.

**Sergey Sekatskii (Lausanne) : Applications of Generalized Li's criterion equivalent to the Riemann hypothesis and Generalized Littlewood Theorem about Contour Integrals involving Logarithm of an Analytical Function to study the Riemann zeta-function zeroes location**

We show that Li's criterion equivalent to the Riemann hypothesis can be generalized in the following way: the sums over Riemann zeta-function zeroes*k*for any real a not equal to ½ are non-negative if and only if the Riemann hypothesis holds true. Applying the generalized Littlewood theorem about contour integrals involving logarithm of an analytical function, we show that this is equivalent to the requirement that all derivatives_{n,a}=Σ_{ρ}(1-(1-((ρ-a)/(ρ+a-1))^{n})*1/((m-1)!)*d*for^{m}/dz^{m}((z-a)^{(m-1)}*ln(χ(z)))*z=1-a*of the Riemann*χ*-function for all real*a<1/2*are non-negative (correspondingly, the same derivatives when*a>1/2*should be non-positive for this; initial Li's criterion is a particular case of*a=1*or*a=0*). The shortcoming related with the impossibility to use the point*z=1/2*for the evaluation of derivatives is removed analyzing quite recent Voros' criterion : we have established the polynomials*P*over*(z-1/2)*such that the derivatives at^{n}*z=1/2*of*P*ln(χ(z))*are non-negative if and only if the Riemann hypothesis holds true. In the second part of the talk we present the first application of this same generalized Li's criterion and prove the following statement: for any positive integer*n*there is such real value of*a*(depending on_{n}*n*) that for all and*a<a*, inequality_{n}*1/((m-1)!)*d*is greater than or equal to^{m}/dz^{m}((z-a)^{(m-1)}*ln(χ(z)))*0*does hold true. Finally, other applications of the generalized Littlewood theorem to Riemann function studies are given by establishing an infinite number of equalities involving integrals of the logarithm of the Riemann zeta-function equivalent to the Riemann hypothesis. In particular, we show that all earlier known criteria of this kind, viz. Wang, Volchkov, Balazard-Saias-Yor and Merlini integral equalities, are certain particular cases of the general approach proposed. Here is the bibliography.

**Alexey Zaytsev (Kaliningrad) : Coefficients of Zeta functions of the second Garcia-Stichtenoth tower as polynomial functions**

Let*Τ=(T*be a tower of function fields over a finite field. Then there exits a sequence of zeta functions (orn)_ _{n ≥ 1}*L*-polynomials)*(Z*(or (_{n}(t))_{n≥ 1}*L*) attached to it. It is well know that_{n}(t))_{n≥ 1}*L*-polynomial*L*is a polynomial of degree_{n}(t)*2g*, where*g*is genus of*T*. It turns out that if a tower is recursive then we can attach a certain graph to it. Using this graph one can express the coefficients of_{n}*L*-polynomials of function field*T*as functions in_{n}*n*. We also formulate some conjecture on coefficients of*L*-polynomial for particular example of towers.