novelty-meter.md

Lay a grid of n equal cells on the unit interval for every n up to N and stack the layers. A reduced fraction a/b is drawn first by layer b, so layer n lights phi(n) nodes for the first time: the novelty of scale n is Euler's totient, and its Dirichlet series is zeta(s-1)/zeta(s). Read the novelty of all scales near 1/y through a window f supported on [1, 2], as S_f(y) = sum_n phi(n) f(ny); the main term is (6/pi^2) F(2) y^-2 with F the Mellin transform of f, and the scaled error E_f(y) = y^2 S_f(y) - (6/pi^2) F(2) is the meter. A theorem of Verjovsky, read here at source, says that the meter decays like y^(3/2 - eps) for every eps and every smooth f if and only if the Riemann hypothesis holds, while for the indicator of [1, 2] the exponent is 1 and no better. This paper assembles that equivalence from the Mellin identity with both inputs named, proves the sharp exponent from the jump of the totient sum at a prime, writes the smoothed error under the hypothesis as a sum over the zeros, and reads the meter: on y = 2^-j for j from 8 to 23.5 with the totients sieved to 3 * 10^7, the fitted exponents in q = y^2 are 0.5023 for the indicator and 0.7498 and 0.7471 for a C^2 and a C^infinity bump, against 1/2 and 3/4, and the residue sum over the first 138 zeros reproduces the smooth bump's error over the whole grid to a relative 2.0e-6. The stack hears the zeros. It cannot tell where they are: on any finite range the smoothed error is a rendering of the first zeros, a zero off the line at a height already checked would be invisible by the same decay that makes the sum converge, and the meter is the hypothesis restated, not a way in.

Introduction

Draw the lines x = a/n for a = 0, ..., n on the unit interval, once for every n from 1 to N, with the ink thin enough that lines add. That is the stack, and the Farey page says what it draws: the Farey fractions of denominator at most N. Layer n adds the fractions of denominator exactly n, phi(n) of them in (0, 1], so a prime scale adds n - 1 new nodes and a composite one fewer. That count is the novelty of the scale, and it is the only arithmetic this paper needs.

The question is how regularly the novelty grows. Summed sharply, sum_(n <= x) phi(n) is 3x^2/pi^2 up to an error that jumps by about 0.39 p at every prime p, so no bound smaller than x is possible. Summed through a smooth window, the jumps average out and the error becomes a sum of waves, one per zero of the Riemann zeta function, the wave of rho = 1/2 + i gamma oscillating like cos(gamma log y) with an amplitude falling like y^(3/2). The figure above is that sum: the dots are the measured error of one smooth window scaled by y^(3/2) at 201 values of y between 2^-8 and 2^-20.5, the line through them is the prediction of the first 29 zeros, computed from their heights, two values of zeta at each and the window's Mellin transform there, with nothing fitted, and below it the sharp window's error at the same scales, scaled by y, is a cloud.

The decay exponent of the smoothed error is a meter for the Riemann hypothesis. If the hypothesis holds, every wave has amplitude y^(3/2) and the error is O(y^(3/2 - eps)); if it fails, a zero off the line contributes a wave that decays more slowly, and the rate of decay over all smooth windows locates the rightmost zero. That is a theorem of Verjovsky (1994), stated in Section 3 as at source and translated into this paper's variable. Section 4 assembles the equivalence from the Mellin transform and derives the wave sum; from the rate to the zeros needs nothing beyond absolute convergence and Weierstrass, the other way needs one input from the literature, the bound on 1/zeta right of the critical line under the hypothesis with the horizontal lines the contour closes on, cited at the page that proves it. Section 5 proves the sharp exponent from the prime jump, Section 6 reads the meter, every number a Fact with its finite domain and the study that prints it, and Section 7 says what the reading is worth.

Definitions

Definition 2.1 (novelty). The novelty of scale n >= 1 is phi(n), the number of reduced fractions in (0, 1] with denominator n. The novelty series is sum_n phi(n) n^-s.

Lemma 2.2. For Re s > 2, sum_n phi(n) n^-s = zeta(s-1)/zeta(s).

Proof. sum_(d | n) phi(d) = n, since every a/n with 1 <= a <= n reduces to exactly one fraction of denominator d | n. Multiplying the two absolutely convergent series, zeta(s) sum_n phi(n) n^-s = sum_n (sum_(d | n) phi(d)) n^-s = sum_n n^(1-s) = zeta(s-1). □

Definition 2.3 (window and meter). A window is a bounded measurable f on (0, infinity) supported in [1, 2]. Its Mellin transform F(s) = int_0^infinity f(u) u^(s-1) du is entire. The smoothed novelty is S_f(y) = sum_n phi(n) f(ny), a finite sum over the scales n between 1/y and 2/y, and the meter is

E_f(y) = y^2 S_f(y) - (6/pi^2) F(2)

read as a power of y as y -> 0, or of q = y^2, which is the variable of the source. An exponent a in q is 2a in y.

Definition 2.4 (the three windows). The sharp window is the indicator of [1, 2], F(s) = (2^s - 1)/s, F(2) = 3/2. The C^2 window is 64 (u-1)^3 (2-u)^3 on (1, 2), F(2) = 24/35. The C^infinity window is exp(4 - 1/((u-1)(2-u))) on (1, 2), F(2) = 0.575725895994 by a 2000-node Gauss-Legendre rule (lab/py/smoothed-novelty, main). The constants 64 and 4 make both bumps peak at 1 at u = 3/2.

The theorem at source

Verjovsky, Discrete measures and the Riemann hypothesis, Kodai Math. J. 17 (1994), defines on page 596, for y > 0, the measure m_y(f) = sum_(n in N) y phi(n) f(y^(1/2) n) on functions of compact support in the positive reals, and on page 597 m_0(f) = int_0^infinity (6/pi^2) u f(u) du. Its y is this paper's q, so m_q(f) = y^2 S_f(y) at q = y^2, m_0(f) = (6/pi^2) F(2), and E_f(y) = m_q(f) - m_0(f). Page 597 states, and the paper proves:

Theorem A. For every f in C_c^0(R*), m_y(f) = (6/pi^2) int_0^infinity u f(u) du + O(y^(1/2) log y) as y -> 0.

Theorem B. (1) The Riemann hypothesis is true if and only if for every f in C_c^r(R*), 2 <= r <= infinity, m_y(f) = m_0(f) + o(y^(3/4 - eps)) as y -> 0, for all eps > 0. (2) For alpha in (1/2, 3/4), m_y(f) = m_0(f) + o(y^(alpha - eps)) for all f in C_c^2(R*) and all eps > 0 holds if and only if zeta has no zeroes in the half-plane Re s > 2(1 - alpha). (3) If f is the characteristic function of an interval then limsup_(y -> 0) y^-alpha |m_y(f) - m_0(f)| = infinity if alpha > 1/2; hence 1/2 is the best possible exponent of the error for some nonsmooth functions. Part (4), on one specific window and zeros near Re s = 1, is not used here.

The same page defines m_y of the characteristic function of [a, b] as the sum over a y^(-1/2) <= n <= b y^(-1/2), both ends included, the convention of Definition 2.4, and page 598 derives Theorem A for it from sum_(n <= x) phi(n) = 3x^2/pi^2 + O(x log x), attributed there to Mertens (1874). The 2017 text of the author's 1993 Pitman notes, arXiv:1711.03593, restates the smooth case as its Theorem 5.1 in the variable y = q^(1/2): for all g in C_0^infinity(R*) the error is o(y), and o(y^(3/2 - eps)) for all eps > 0 if and only if the Riemann hypothesis holds, the proof referred to the Kodai article. Both texts were read at source. In this paper's variable the smooth meter sits at 3/2 - eps in y exactly when the hypothesis holds, and the sharp meter at 1 in y unconditionally and sharply.

The meter is a Riemann hypothesis meter

Theorem 4.1 (the Mellin identity). Let f be a C^2 window and c > 2. Then

S_f(y) = (1/(2 pi i)) int_(Re s = c) F(s) (zeta(s-1)/zeta(s)) y^-s ds

and the residue of the integrand at s = 2 is (6/pi^2) F(2) y^-2.

Proof. Two integrations by parts, the boundary terms vanishing because f and f' vanish at 1 and 2, give F(s) = (1/(s(s+1))) int f''(u) u^(s+1) du, so |F(s)| <= 2^(Re s + 1) ||f''||_1 / (|s| |s+1|) uniformly on vertical strips. Mellin inversion therefore holds pointwise with an absolutely convergent integral, f(u) = (1/(2 pi i)) int_(Re s = c) F(s) u^-s ds. Put u = ny, multiply by phi(n) and sum; sum_n phi(n) n^-c converges for c > 2 and int |F(s)| |ds| is finite, so the sum and the integral exchange, and Lemma 2.2 gives the display. At s = 2 the factor zeta(s-1) has a simple pole of residue 1 and zeta(2) = pi^2/6. □

Theorem 4.2 (the hypothesis gives the rate). Assume the Riemann hypothesis. Then for every C^2 window f and every eps > 0, E_f(y) = O(y^(3/2 - eps)).

Proof. Move the line of Theorem 4.1 from Re s = c to Re s = 1/2 + eps. Two facts about zeta under the hypothesis are needed, and both are taken from Hu, Kaneko, Martin and Schildkraut (2023), where they are proved for any Dedekind zeta and, at K = Q, are Corollary 13.16 and Theorem 13.22 of Montgomery and Vaughan (2007) by that paper's own attribution. Its Lemma 5.4, with tau = |t| + 4 and |t| >= 1, bounds |log zeta(sigma + it)| by log(1/(sigma - 1)) + O(sigma - 1) on 1 + 1/log log tau <= sigma <= 3/2, by log log log tau + O(1) on 1 - 1/log log tau <= sigma <= 1 + 1/log log tau, and by log(1/(1 - sigma)) + O((log tau)^(2 - 2 sigma) / ((1 - sigma) log log tau)) on 1/2 + 1/log log tau <= sigma <= 1 - 1/log log tau. The third range covers 1/2 + eps <= sigma <= 1 - 1/log log tau once tau is large, the second and first cover sigma up to 3/2, and beyond 3/2 the Euler product gives |1/zeta(s)| <= zeta(3/2); together they exponentiate to 1/zeta(s) << tau^delta for every delta > 0, uniformly on Re s >= 1/2 + eps. Its Lemma 2.4 gives, for each integer n >= 4, a height T_n in [n, n + 1) with |zeta(sigma + i T_n)| >= exp(-C log n / log log n) for -1 <= sigma <= 2. On the new line zeta(s - 1) is O(|t|^(1 - eps)) by the functional equation, F(s) = O(|s|^-2) by the bound in the proof of Theorem 4.1, and 1/zeta(s) = O(|t|^delta), so the integrand is O(|t|^(-1 - eps + delta)) and the integral converges absolutely for delta < eps, giving O(y^(-1/2 - eps)). On the horizontal segments at height T_n between the two lines the integrand is O(T_n^(-1 + o(1)) y^-c) and vanishes as n -> infinity. Under the hypothesis no zero of zeta lies in Re s > 1/2 + eps, so the only residue crossed is the one at s = 2, and S_f(y) = (6/pi^2) F(2) y^-2 + O(y^(-1/2 - eps)). Multiply by y^2. □

Theorem 4.3 (the rate gives the hypothesis). Fix eps > 0. If E_f(y) = O(y^(3/2 - eps)) for every C^2 window f, then zeta has no zero in Re s > 1/2 + eps. If it holds for every eps > 0, the Riemann hypothesis holds.

Proof. S_f(y) = 0 for y > 2, since f(ny) = 0 once ny > 2. For Re s > 2, by absolute convergence, int_0^2 S_f(y) y^(s-1) dy = sum_n phi(n) int_0^infinity f(ny) y^(s-1) dy = F(s) zeta(s-1)/zeta(s). Writing S_f(y) = ((6/pi^2) F(2) + E_f(y)) y^-2, the left side is

(6/pi^2) F(2) 2^(s-2)/(s-2) + int_0^2 E_f(y) y^(s-3) dy .

Under the rate the integrand is O(y^(Re s - 3/2 - eps)) near 0 and bounded near 2, so the integral converges absolutely and locally uniformly on Re s > 1/2 + eps and the display is holomorphic there except for the simple pole at s = 2. So F(s) zeta(s-1)/zeta(s) continues holomorphically to Re s > 1/2 + eps away from s = 2. Let rho be a zero of zeta there. Since zeta has no zeros on Re s >= 1, Re rho < 1, and Re(rho - 1) lies in (-1/2, 0), where zeta has no zeros; so zeta(rho - 1) != 0, and holomorphy at rho forces F(rho) = 0. Now u^k f(u) is again a C^2 window for every integer k >= 0, with Mellin transform F(s + k), so the same argument gives F(rho + k) = 0 for every k: the continuous function g(u) = f(u) u^(rho - 1) on [1, 2] is orthogonal to every polynomial. By Weierstrass, polynomials are dense in C[1, 2], so g is orthogonal to its own conjugate and g = 0, whence f = 0. Taking any nonzero window gives the contradiction. The second sentence follows by letting eps -> 0. □

Theorems 4.2 and 4.3 together are Theorem B part (1) at r = 2 for windows on [1, 2], and the two halves do not cost the same: the backward half is elementary, the forward half rests on a bound on 1/zeta that is itself a consequence of the hypothesis.

Theorem 4.4 (the wave sum). Assume the Riemann hypothesis and that the nontrivial zeros of zeta are simple, and let f be a C^infinity window, or a C^2 window whose transform is O(|s|^-4), as the C^2 bump of Definition 2.4 is. Then for every 0 < delta < 1/2,

E_f(y) = sum_rho F(rho) (zeta(rho - 1)/zeta'(rho)) y^(2 - rho) + O(y^(3/2 + delta)) ,

the sum over the nontrivial zeros taken as residues in the order of the heights T_n of Theorem 4.2. Each term has modulus |c_rho| y^(2 - Re rho) with c_rho = F(rho) zeta(rho - 1)/zeta'(rho), which is |c_rho| y^(3/2) on the line; a zero of real part beta would contribute a term of exponent 1 - beta/2 in q, which is 3/4 on the line and smaller at the member of a pair beta, 1 - beta that sits right of it.

Proof. Integrating by parts k times, F(s) = O(|s|^-k) for every k when f is C^infinity, since every derivative vanishes at the endpoints; for the C^2 bump three integrations by parts and one more boundary term give O(|s|^-4), f''' being its first derivative not vanishing at the endpoints. Shift the line of Theorem 4.1 to Re s = 1/2 - delta, crossing the pole at s = 2 and the zeros. On the new line zeta(s - 1) = O(|t|^(1 + delta)) and 1/zeta(s) = O(|t|^(delta')) for every delta' > 0, the latter by the functional equation and the bound of Theorem 4.2 at Re s = 1/2 + delta, so against F = O(|s|^-4) the integrand is O(|t|^(-3 + delta + delta')), the integral converges absolutely and is O(y^(-1/2 + delta)). The horizontal segments at the heights T_n vanish as before, and the residue at a simple zero rho is F(rho) zeta(rho - 1) y^-rho / zeta'(rho). Multiply by y^2. The exponent count is |y^(2 - rho)| = y^(2 - Re rho) and q = y^2. □

The wave of rho = 1/2 + i gamma is 2 Re(c_rho y^(3/2 - i gamma)) = 2 |c_rho| y^(3/2) cos(gamma log y - arg c_rho), one cosine in log y per zero at the zero's height as frequency: the top panel of the figure. What smoothness buys is not the shift, which the C^2 bump already admits, but the speed at which the coefficients die: for the C^infinity window F(rho) falls faster than any power of gamma, so a few dozen zeros are the whole error, while for the C^2 bump they fall like a power and the sum converges slowly (Fact 6.2). For the indicator F(s) = (2^s - 1)/s is only O(1/|t|), the contour cannot cross the line at all, and the sharp exponent comes from somewhere else.

The sharp cutoff

Theorem 5.1. Let R(x) = sum_(n <= x) phi(n) - 3x^2/pi^2. Then R(x) = O(x log x), and R(x) = Omega(x): across every prime p it jumps by phi(p) - 3(2p - 1)/pi^2 = (1 - 6/pi^2) p - 1 + 3/pi^2 > 0.39 p - 1, so one of |R(p - 1)| and |R(p)| exceeds 0.19 p - 1.

Proof. The identity sum_(d | n) phi(d) = n of Lemma 2.2 inverts to phi = mu * id, so sum_(n <= x) phi(n) = sum_(d <= x) mu(d) T(floor(x/d)) with T(m) = m(m+1)/2. Since T(floor(x/d)) = x^2/(2d^2) + O(x/d) and sum_(d <= x) mu(d)/d^2 = 6/pi^2 + O(1/x), the sum is 3x^2/pi^2 + O(x) + O(x sum_(d <= x) 1/d) = 3x^2/pi^2 + O(x log x). The jump is the definition of R at p - 1 and at p, with phi(p) = p - 1. □

The upper bound is Mertens' (1874), recalled on page 598 of the source; the lower bound is the only part the meter feels.

Corollary 5.2 (the sharp meter). For the indicator f of [1, 2], E_f(y) = O(y log(1/y)) and E_f(y) = Omega(y): at y = 2/p for an odd prime p the meter jumps by y^2 phi(p) = (1 - 1/p) 2y, so the exponent of the sharp meter is 1 in y and 1/2 in q, unconditionally, and it is attained.

Proof. S_f(y) = sum_(1/y <= n <= 2/y) phi(n), so E_f(y) = y^2 (R(floor(2/y)) - R(ceil(1/y) - 1)) + y^2 (3/pi^2)(floor(2/y)^2 - (ceil(1/y) - 1)^2) - 9/pi^2, and the last two terms cancel to O(y), which with Theorem 5.1 gives the upper bound. As y decreases through 2/p the upper end floor(2/y) steps from p - 1 to p while ceil(1/y) = (p+1)/2 does not move, so S_f jumps by phi(p) and E_f by y^2 (p - 1) with y = 2/p; one side of the jump has |E_f| >= (1 - 1/p) y. □

This is Theorem B part (3) at [1, 2], its proof reduced to the prime jump. The sharp meter sits at 1 in y because the primes' jumps are louder than the zeros' waves, which live at 3/2: its exponent says nothing about the zeros. Smoothing is the whole difference between the two readings, and the difference is the quarter.

The meter read

All numbers in this section are printed by uv run python research/lab/py/smoothed-novelty/smoothed_novelty.py. The grid is y = 2^-j for j from 8 to 23.5 in steps of 1/16, 249 samples, the largest window summing 1.19 * 10^7 scales; the totients are sieved to 3 * 10^7 and agree with brute-force gcd counts to 2000 and with the Mobius route of Theorem 5.1 at x = 10^6, both exactly (totients, totient_sum_mobius). The summation noise at j = 23.5, pairwise against compensated summation, is 5.6e-17 against |E_f| = 5.4e-12.

Fact 6.1 (the slopes). An octave is the set of samples with j in [k, k + 1), 16 of them for k from 8 to 22 and the 9 of [23, 23.5] for the last, each placed at j = k + 1/2. Fit log of the root mean square of E_f over each octave against log q over the 16 octaves, and over the lower and upper eight. The slopes in q read:

Table 1. Exponents of the meter in q = y^2, against 1/2 for the sharp window and 3/4 for the smooth ones. Residuals are root mean square in log units; the eight-octave windows carry residuals between 0.065 and 0.169, and no two windows of one row differ by more than 0.05. Domain: the 249-sample grid above. Script: slopes. No exponent is claimed beyond that window.

The smoothed slopes do not read 1/2: the smoothing gains the quarter that Theorem B states, and the C^2 bump gains all of it, as part (1) says it must at r = 2.

Fact 6.2 (the wave sum against the first 138 zeros). The 138 zeros of zeta to height 300, with zeta(rho - 1) and zeta'(rho) at each, are read from PARI in one call (zeros_from_pari; lfunzeros, zeta, lfun at derivative order 1, 30 digits). For the C^infinity bump the residue sum of Theorem 4.4, 2 Re sum_rho c_rho y^(2 - rho), reproduces the measured E_f over the whole grid to a relative 2.0e-6 in the maximum norm; one zero alone gives 0.50, ten 2.7e-2, thirty 2.0e-3, and the coefficients |c_rho| fall from 1.879e-1 at the first zero to 1.475e-7 at the 138th. The scaled amplitude |E_f|/y^(3/2) lies in [1.3e-4, 0.558] over the grid against 2 sum |c_rho| = 0.755 over the same zeros. For the C^2 bump the same sum stops at 2.3e-4, its coefficients falling only like a power, the printed least-squares power against gamma being -3.22 on the 138. Domain: the 249-sample grid, zeros to height 300. Script: main, mellin_cinf, mellin_c2.

Fact 6.3 (the prime jump). R(p - 1) = 108941.6 and R(p) = 501014.9 at p = 1000003, a jump of 392073.4; R(p - 1) = 1011363.8 and R(p) = 4932099.6 at p = 10000019, a jump of 3920735.7; both equal phi(p) - 3(2p - 1)/pi^2 to the printed digit. Script: main, SHARP CUTOFF.

The figure is the same computation at a smaller height, run by its own binary, paper-novelty-meter: phi sieved to 3 * 10^6 with mrlynum::lattice::totients, j from 8 to 20.5 in steps of 1/16, 201 samples, so that every window fits under the sieve, the 29 zeros below height 100 from mrlynum::zeta::Line::zeros, zeta(rho - 1) and zeta'(rho) by mrlynum::zeta::Line::pair, checked against the crate's value at 1/2 + 30i and against zeta(-1/2) and zeta'(1/2), and F(rho) by a 4096-node quadrature. It asserts F(2) against 0.575725895994, |c_rho| at the first and tenth zeros against 1.879e-1 and 4.286e-3, and the 29-zero sum against the drawn samples within one percent of their peak. The top panel is E_f(y)/y^(3/2) for the C^infinity bump, dots, with the 29-zero sum as the line; the bottom panel is E_f(y)/y for the indicator, dots alone; both run from j = 8 on the left to 20.5 on the right.

What the meter does not do

Three things, said plainly. First, the equivalence of Section 4 is exactly as hard as the Riemann hypothesis: it is the hypothesis rewritten in y, and the same wall is reached by the Farey discrepancy on the Farey page and by the Gaussian Franel identity on the stack page. Second, a finite reading cannot certify or refute. Fact 6.2 says the smoothed error on the whole grid is the first 138 zeros to six digits, and the reason is Theorem 4.4: the Mellin transform of a smooth window kills the coefficients of high zeros faster than any power, so a zero off the line at a height where the hypothesis has already been checked would contribute a wave too small to see, by the same decay that makes the sum converge. The smoothed novelty meter is not a route to the hypothesis. Third, the slopes of Table 1 are fits on sixteen octaves with residuals of a tenth in log, and the residual is not noise but the waves themselves, whose octave averages beat against one another; no exponent is claimed beyond the window read.

Open problems

Theorem 4.4 is written under the hypothesis and for simple zeros. A multiple zero contributes the residue of the same integrand and changes nothing else. Dropping the hypothesis is not so cheap: the horizontal lines of the contour come from Lemma 2.4 of Hu, Kaneko, Martin and Schildkraut, which assumes it, and the unconditional substitute, 1/zeta << exp(C log^2 T) on chosen heights (Titchmarsh 1986, chapter 9), beats every polynomial decay of F, so a general C^infinity window does not obviously survive; a window whose transform decays like exp(-c sqrt(|t|)) would, and whether the bump used here does is not checked. Neither version is written here. Two questions are left open beyond that: which window of a given support makes a given zero's coefficient |F(rho) zeta(rho - 1)/zeta'(rho)| largest relative to the rest, so that the meter hears that zero best, nothing here optimising f; and whether the octave-to-octave residual of the smooth fits in Table 1, 0.092 and 0.115, is the beat of the first few waves alone, which it should be by Fact 6.2 and which has not been checked. Neither is claimed.

Reproducibility

One study, lab/py/smoothed-novelty, prints every number of Section 6: uv run python research/lab/py/smoothed-novelty/smoothed_novelty.py from the repository root, one core, about fifteen seconds, needing numpy and gp on the path, taking no arguments, reading and writing no file; its README names the witness of every printed line. The figure is bash scripts/figures.sh paper-novelty-meter, under a second a theme, and every quantity it draws is asserted inside the binary as listed in Section 6, so a wrong sieve, zero, zeta or Mellin transform stops the press.

References

• Verjovsky 1994, Discrete measures and the Riemann hypothesis, Kodai Math. J. 17, no. 3, 596-608. doi.org/10.2996/kmj/1138040054
• Verjovsky 2017, Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold, the updated text of the 1993 Pitman Research Notes article, Theorem 5.1. arxiv.org/abs/1711.03593
• Hu, Kaneko, Martin and Schildkraut 2023, On a Mertens-type conjecture for number fields, Math. Proc. Cambridge Philos. Soc., Lemma 5.4 and Lemma 2.4. arxiv.org/abs/2109.06665
• Montgomery and Vaughan 2007, Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics 97, Corollary 13.16 and Theorem 13.22. cambridge.org
• Titchmarsh 1986, The Theory of the Riemann Zeta-Function, second edition revised by D. R. Heath-Brown, Clarendon Press, chapter 9. sites.math.rutgers.edu
• Mertens 1874, Ueber einige asymptotische Gesetze der Zahlentheorie, J. reine angew. Math. 77, 289-338. doi.org/10.1515/crll.1874.77.289