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        Questions tagged [analytic-number-theory]

        A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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        17 views

        Supremum of certain modified zeta functions at 1

        Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
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        votes
        0answers
        56 views

        Linear exponential sum with gcd

        The sum $$\sum _{d,d'\leq D}\sum _{h,h'=1}^q(h,q)e\left (\frac {dh+d'h'}{q}\right )$$ is easily seen to be $$\ll q^{2+\epsilon }+D^2.$$ Indeed with a standard estimate for a linear exponential sum it ...
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        vote
        0answers
        52 views

        Maximum number of bounded primitive integer points in a zero-dimensional system

        Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
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        vote
        0answers
        56 views

        What is the probability of 'yes' to this likely $coNP$ problem?

        Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$. Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
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        votes
        2answers
        93 views

        Real non trivial zeros of Dirichlet L-functions

        When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
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        votes
        3answers
        4k views

        Why is so much work done on numerical verification of the Riemann Hypothesis?

        I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ...
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        votes
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        71 views

        Subgroup of the symmetry group of $Zer(\zeta)$ preserving multiplicity

        Let $Zer(\zeta)$ denote the multiset of the non trivial zeros of the Riemann zeta function counted with multiplicity and $G$ the group of isometries of the complex plane preserving this multiset ...
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        votes
        1answer
        180 views

        Frequency of digits in powers of $2, 3, 5$ and $7$

        For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example, $$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$ Similarly, define the ...
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        votes
        1answer
        116 views

        On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function

        Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime. Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
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        votes
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        185 views

        On Robin's inequality and the zeros of the Riemann zeta function

        Let $\zeta$ denote the Riemann zeta function. By an argument of Robin http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $\zeta(\rho)=0$ for some $\rho$ with $\Re(\rho) \in (1/2, 1/2 + \beta]$,...
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        votes
        1answer
        298 views

        Yet another question on sums of the reciprocals of the primes

        I recall reading once that the sum $$\sum_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$ is less than $4$. Does anybody here know what the ultimate source of this claim is? Please, let me ...
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        0answers
        121 views

        A strengthening of Dirichlet prime number theorem

        Dirichlet Theorem on arithmetic progression states that the sequence $\{a+kd\}_{k=1}^{\infty}$ contains infinitely many primes when $(a,d)=1$. In other words if we let $A=\{a+kd\}_{k=1}^{\infty}$ and ...
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        170 views

        Prime character sums

        Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
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        0answers
        137 views

        Riemann Explicit Formula

        I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula: $$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
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        votes
        1answer
        318 views

        Sum over characters

        Take $x>0$ large, $t\in \mathbb R$, $q\in \mathbb N$ and a non-principal character $\chi $ mod $q$. If you want, take $t\leq x$. How do I bound \[ \sum _{n\leq x}\frac {\chi (n)}{n^{it}}?\] My ...

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