# 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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### Modified Gauss Sum when the characters have different period

Let $\chi$ be a Dirichlet character mod q, and \begin{eqnarray} t(\chi)=\sum_{n=1}^{q}\chi(n)e(\frac{n}{2q}). \end{eqnarray} Do we have a bound or formula for $t(\chi)$ similar to that of the usual ...
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### Generators of the ideal class group

Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following: Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
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### Error function of the second moment of the divisor function

It is easy to show that the second moment of the divisor function has asymptotics: $$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$ Where $P$ is some polynomial and that: $$E_2 = o(x)$$ Previously, ...
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### Electrostatic potential energy of point-charges at primes up to $x$

Given a positive real (or integral) number $x$ we consider the electrostatic potential energy of equal point charges at all primes up to $x$ given by $$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$ ...
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### Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$

It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$ for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
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### Density of a set of numbers whose prime factors are defined by congruences

Let $S$ be the set of positive integers not divisible by $3$ where if $p$ is a prime factor of $n \in S$ and $p \equiv 1\bmod 3$ then $p^2$ does not divide $n$, but if $p\equiv2 \bmod 3$ then $p^2$ ...
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### Number of zeros of the zeta function along horizontal lines

Are there any known results about the number of zeros of the zeta function along horizontal lines of the complex plane? The Riemann hypothesis states that for any such line the number is at most 1, ...
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### Reference request - Pillai-Selberg Theorem

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
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### A strictly increasing, analytic function that goes through key points of the iterated logarithm?

Is it possible to create a function $f(x)$ that: is strictly increasing (at least for $x>0$) is real analytic goes through all the points where the iterated logarithm would increment value? i.e. [...
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### What is the lattice point distribution over binary quadratic forms?

Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$. For simplicity, we keep things only on quadrant I of the ...
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### Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
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### Real part of the Riemann zeta function

Consider the real part of the Riemann zeta function on the critical line. Are there any results for the number of zeros of this real function in the interval [0,T]?
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### Large prime divisors of values of a polynomial, in a given residue class

Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
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### Upper bound on sum of Lambda(n) over short interval

I am looking for a bound of type $$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$ (or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
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This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function: \xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\... • 4,179 2 votes 0 answers 131 views ### Analyticity of unramifed part of Rankin-Selberg L-functions on \Re(s)=1 I have only a little knowledge about automorphic representations and L-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ... • 653 6 votes 0 answers 266 views ### Approximating \zeta'/\zeta (and its derivatives) by a finite sum Let A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}, where r\geq 0. (We can consider r=0 first for simplicity.) Say I want to approximate A(s) for s=1+it by a finite sum - preferably a ... • 19.3k 0 votes 0 answers 193 views ### Does there exist an L-function for any subset of \mathbb{N}? Consider the following prime sum: \begin{aligned} \sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}} \end{aligned} whose spikes appear at the Riemann \zeta zeros as shown here. Taking these detected spikes (... • 1,893 1 vote 0 answers 261 views ### On fifth powers forming a Sidon set We call a set of natural numbers \mathcal S to be a Sidon Set if a+b=c+d for a,b,c,d\in \mathcal S implies \{a,b\}=\{c,d\}. In other words, all pairwise sums are distinct. Erdős conjectured ... • 711 0 votes 1 answer 229 views ### Is there any use of logarithmic derivatives of modular forms? Does taking the logarithmic derivative of a modular form have any uses, such as identifying patterns in its coefficients or possible zeros of its corresponding L function? 27 votes 1 answer 2k views ### Is every real number in [0,1] a product of three (or more) Cantor set's numbers? It is well known that every number x in the unit interval [0,1] is the arithmetic mean of two elements of the (triadic) Cantor set C. The way to see it I like the most: the Cantor set is the ... • 56k 5 votes 0 answers 125 views ### Taking integer values of a sequence of Beurling primes Let P=(p_j)_{j=1}^\infty be an increasing sequence of real numbers with 1<p_1 and \lim_{j\to\infty}p_j=\infty. As mentioned in [1], Beurling proved that if the multiplicative group N_P ... 2 votes 1 answer 221 views ### '\times' or '\otimes' when writing L-functions? Recently, I came across the Langlands correspondence theorem, there is the following line:L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$where \sigma and \tau are ... • 143 18 votes 1 answer 1k views ### Does summing divergent series using cutoff functions give consistent results? One way to try to give a value S to a divergent series \sum_{n=1}^\infty a_n is with a smooth cutoff function:$$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$where \... 2 votes 0 answers 179 views ### Can all modular forms be written as Eta Quotients? I have been going through a couple of introductory courses in modular forms and am quite curious whether all modular forms can be written as eta quotients of the Dedekind eta function? 4 votes 2 answers 578 views ### Computing hypergeometric function at 1 I'm looking to compute$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr 1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$for m,k > 0 are positive integers and 0 < k < m. I'm wondering if ... • 259 1 vote 0 answers 103 views ### Behavior of Dirichlet L-functions at the edge of the critical strip Given a Dirichlet L-function L(\chi, s) of a primitive character \chi, what is the asymptotic behavior of L(\chi, 1+it) for real t? I am looking for as many answers for the same question. This ... 1 vote 0 answers 156 views ### Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane? I would like to understand a little bit about how to interpret and construct 1-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ... 8 votes 1 answer 627 views ### Does this partial sum over primes spike at all zeta zeros? Below is a plot of \exp \sum _p^x -\frac{\cos \left(x \log \ p\right)}{\sqrt{p}}, where p runs over the primes, and the x-values of the Riemann \zeta zeros are marked with dashed lines: Below ... • 1,893 0 votes 0 answers 69 views ### Decrease of (1/\zeta)^{(r)}(\sigma + i T) as \sigma\to -\infty? What is a standard reference for the simple fact that, for T fixed and \sigma\to -\infty, every derivative |(1/\zeta)^{(r)}(\sigma+i T)| of the Riemann zeta function decreases faster than any ... • 19.3k 2 votes 2 answers 279 views ### L^1 norm for a product of cosines Let k be an integer and consider the function$$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).  I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
I am attempting to understand the behavior of Hurwitz zeta functions and for what $a$ do they have an analytic continuation. Is it possible to write any Hurwitz zeta as an Euler product or are there ...